Description: A set equinumerous to ordinal 2 is a pair. (Contributed by Mario Carneiro, 5-Jan-2016)
Ref | Expression | ||
---|---|---|---|
Assertion | en2 | ⊢ ( 𝐴 ≈ 2o → ∃ 𝑥 ∃ 𝑦 𝐴 = { 𝑥 , 𝑦 } ) |
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 | ⊢ ( 𝑥 ∈ 𝐴 → ( ( 𝐴 ∖ { 𝑥 } ) = { 𝑦 } → 𝐴 = { 𝑥 , 𝑦 } ) ) |
8 | 7 | eximdv | ⊢ ( 𝑥 ∈ 𝐴 → ( ∃ 𝑦 ( 𝐴 ∖ { 𝑥 } ) = { 𝑦 } → ∃ 𝑦 𝐴 = { 𝑥 , 𝑦 } ) ) |
9 | 2 3 5 8 | enp1i | ⊢ ( 𝐴 ≈ 2o → ∃ 𝑥 ∃ 𝑦 𝐴 = { 𝑥 , 𝑦 } ) |