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Mirrors > Home > MPE Home > Th. List > en2 | Structured version Visualization version GIF version |
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 8473 | . . 3 ⊢ 1o ∈ On | |
2 | 1 | onordi 6465 | . 2 ⊢ Ord 1o |
3 | df-2o 8462 | . 2 ⊢ 2o = suc 1o | |
4 | en1 9017 | . . 3 ⊢ ((𝐴 ∖ {𝑥}) ≈ 1o ↔ ∃𝑦(𝐴 ∖ {𝑥}) = {𝑦}) | |
5 | 4 | biimpi 215 | . 2 ⊢ ((𝐴 ∖ {𝑥}) ≈ 1o → ∃𝑦(𝐴 ∖ {𝑥}) = {𝑦}) |
6 | df-pr 4623 | . . . 4 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦}) | |
7 | 6 | enp1ilem 9274 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∖ {𝑥}) = {𝑦} → 𝐴 = {𝑥, 𝑦})) |
8 | 7 | eximdv 1912 | . 2 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦(𝐴 ∖ {𝑥}) = {𝑦} → ∃𝑦 𝐴 = {𝑥, 𝑦})) |
9 | 2, 3, 5, 8 | enp1i 9275 | 1 ⊢ (𝐴 ≈ 2o → ∃𝑥∃𝑦 𝐴 = {𝑥, 𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∖ cdif 3937 {csn 4620 {cpr 4622 class class class wbr 5138 1oc1o 8454 2oc2o 8455 ≈ cen 8932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-ord 6357 df-on 6358 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-1o 8461 df-2o 8462 df-en 8936 |
This theorem is referenced by: en3 9278 hash2pr 14427 pmtrrn2 19370 trsp2cyc 32750 en2pr 42787 pr2cv 42788 |
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