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| Description: A set equinumerous to ordinal 2 is a pair. (Contributed by Mario Carneiro, 5-Jan-2016.) | 
| Ref | Expression | 
|---|---|
| en2 | ⊢ (𝐴 ≈ 2o → ∃𝑥∃𝑦 𝐴 = {𝑥, 𝑦}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 1on 8518 | . . 3 ⊢ 1o ∈ On | |
| 2 | 1 | onordi 6495 | . 2 ⊢ Ord 1o | 
| 3 | df-2o 8507 | . 2 ⊢ 2o = suc 1o | |
| 4 | en1 9064 | . . 3 ⊢ ((𝐴 ∖ {𝑥}) ≈ 1o ↔ ∃𝑦(𝐴 ∖ {𝑥}) = {𝑦}) | |
| 5 | 4 | biimpi 216 | . 2 ⊢ ((𝐴 ∖ {𝑥}) ≈ 1o → ∃𝑦(𝐴 ∖ {𝑥}) = {𝑦}) | 
| 6 | df-pr 4629 | . . . 4 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦}) | |
| 7 | 6 | enp1ilem 9312 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∖ {𝑥}) = {𝑦} → 𝐴 = {𝑥, 𝑦})) | 
| 8 | 7 | eximdv 1917 | . 2 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦(𝐴 ∖ {𝑥}) = {𝑦} → ∃𝑦 𝐴 = {𝑥, 𝑦})) | 
| 9 | 2, 3, 5, 8 | enp1i 9313 | 1 ⊢ (𝐴 ≈ 2o → ∃𝑥∃𝑦 𝐴 = {𝑥, 𝑦}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∖ cdif 3948 {csn 4626 {cpr 4628 class class class wbr 5143 1oc1o 8499 2oc2o 8500 ≈ cen 8982 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-1o 8506 df-2o 8507 df-en 8986 | 
| This theorem is referenced by: en3 9316 hash2pr 14508 pmtrrn2 19478 trsp2cyc 33143 en2pr 43560 pr2cv 43561 | 
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