1 + 1 = 2

You know people always say "wah the proof of 1 + 1 = 2 is enough to fill a whole blackboard"

I never really knew the proof actually, throughout my four years in NUS.
But recently I started to pick up Set theory on my own since it's known to be the logical foundation of Mathematics and even though I didn't come across the proof in the lectures notes this proof came up suddenly in my head one day when I was walking to the gym.

First we need define cardinality, which basically means size, in the "counting" sense
Next we need define what is 1 and what is 2 and the empty, null set.

1 is the set containing the null set
2 is the set containing the null set and the set containing the null set.

Assume % is the null set
1 = {%}
2 = {%,{%}}

Next we need define the operation "+", which is a binary, commutative, associative and distributive operation
Next we need define "="

so see that 1 + 1 has the same cardinality as 2.