Metamath Proof Explorer


Theorem en2

Description: A set equinumerous to ordinal 2 is a pair. (Contributed by Mario Carneiro, 5-Jan-2016)

Ref Expression
Assertion en2 ( 𝐴 ≈ 2o → ∃ 𝑥𝑦 𝐴 = { 𝑥 , 𝑦 } )

Proof

Step Hyp Ref Expression
1 1on 1o ∈ On
2 1 onordi Ord 1o
3 df-2o 2o = suc 1o
4 en1 ( ( 𝐴 ∖ { 𝑥 } ) ≈ 1o ↔ ∃ 𝑦 ( 𝐴 ∖ { 𝑥 } ) = { 𝑦 } )
5 4 biimpi ( ( 𝐴 ∖ { 𝑥 } ) ≈ 1o → ∃ 𝑦 ( 𝐴 ∖ { 𝑥 } ) = { 𝑦 } )
6 df-pr { 𝑥 , 𝑦 } = ( { 𝑥 } ∪ { 𝑦 } )
7 6 enp1ilem ( 𝑥𝐴 → ( ( 𝐴 ∖ { 𝑥 } ) = { 𝑦 } → 𝐴 = { 𝑥 , 𝑦 } ) )
8 7 eximdv ( 𝑥𝐴 → ( ∃ 𝑦 ( 𝐴 ∖ { 𝑥 } ) = { 𝑦 } → ∃ 𝑦 𝐴 = { 𝑥 , 𝑦 } ) )
9 2 3 5 8 enp1i ( 𝐴 ≈ 2o → ∃ 𝑥𝑦 𝐴 = { 𝑥 , 𝑦 } )