Metamath Proof Explorer


Theorem en3

Description: A set equinumerous to ordinal 3 is a triple. (Contributed by Mario Carneiro, 5-Jan-2016)

Ref Expression
Assertion en3 ( 𝐴 ≈ 3o → ∃ 𝑥𝑦𝑧 𝐴 = { 𝑥 , 𝑦 , 𝑧 } )

Proof

Step Hyp Ref Expression
1 2on 2o ∈ On
2 1 onordi Ord 2o
3 df-3o 3o = suc 2o
4 en2 ( ( 𝐴 ∖ { 𝑥 } ) ≈ 2o → ∃ 𝑦𝑧 ( 𝐴 ∖ { 𝑥 } ) = { 𝑦 , 𝑧 } )
5 tpass { 𝑥 , 𝑦 , 𝑧 } = ( { 𝑥 } ∪ { 𝑦 , 𝑧 } )
6 5 enp1ilem ( 𝑥𝐴 → ( ( 𝐴 ∖ { 𝑥 } ) = { 𝑦 , 𝑧 } → 𝐴 = { 𝑥 , 𝑦 , 𝑧 } ) )
7 6 2eximdv ( 𝑥𝐴 → ( ∃ 𝑦𝑧 ( 𝐴 ∖ { 𝑥 } ) = { 𝑦 , 𝑧 } → ∃ 𝑦𝑧 𝐴 = { 𝑥 , 𝑦 , 𝑧 } ) )
8 2 3 4 7 enp1i ( 𝐴 ≈ 3o → ∃ 𝑥𝑦𝑧 𝐴 = { 𝑥 , 𝑦 , 𝑧 } )