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Mirrors > Home > MPE Home > Th. List > en3 | Structured version Visualization version GIF version |
Description: A set equinumerous to ordinal 3 is a triple. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Ref | Expression |
---|---|
en3 | ⊢ (𝐴 ≈ 3o → ∃𝑥∃𝑦∃𝑧 𝐴 = {𝑥, 𝑦, 𝑧}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2on 8494 | . . 3 ⊢ 2o ∈ On | |
2 | 1 | onordi 6474 | . 2 ⊢ Ord 2o |
3 | df-3o 8482 | . 2 ⊢ 3o = suc 2o | |
4 | en2 9299 | . 2 ⊢ ((𝐴 ∖ {𝑥}) ≈ 2o → ∃𝑦∃𝑧(𝐴 ∖ {𝑥}) = {𝑦, 𝑧}) | |
5 | tpass 4752 | . . . 4 ⊢ {𝑥, 𝑦, 𝑧} = ({𝑥} ∪ {𝑦, 𝑧}) | |
6 | 5 | enp1ilem 9296 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∖ {𝑥}) = {𝑦, 𝑧} → 𝐴 = {𝑥, 𝑦, 𝑧})) |
7 | 6 | 2eximdv 1915 | . 2 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦∃𝑧(𝐴 ∖ {𝑥}) = {𝑦, 𝑧} → ∃𝑦∃𝑧 𝐴 = {𝑥, 𝑦, 𝑧})) |
8 | 2, 3, 4, 7 | enp1i 9297 | 1 ⊢ (𝐴 ≈ 3o → ∃𝑥∃𝑦∃𝑧 𝐴 = {𝑥, 𝑦, 𝑧}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ∖ cdif 3941 {csn 4624 {cpr 4626 {ctp 4628 class class class wbr 5142 2oc2o 8474 3oc3o 8475 ≈ cen 8954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-1o 8480 df-2o 8481 df-3o 8482 df-en 8958 |
This theorem is referenced by: en4 9301 hash3tr 14477 |
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