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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 8419 | . . 3 ⊢ 2o ∈ On | |
2 | 1 | onordi 6426 | . 2 ⊢ Ord 2o |
3 | df-3o 8407 | . 2 ⊢ 3o = suc 2o | |
4 | en2 9184 | . 2 ⊢ ((𝐴 ∖ {𝑥}) ≈ 2o → ∃𝑦∃𝑧(𝐴 ∖ {𝑥}) = {𝑦, 𝑧}) | |
5 | tpass 4712 | . . . 4 ⊢ {𝑥, 𝑦, 𝑧} = ({𝑥} ∪ {𝑦, 𝑧}) | |
6 | 5 | enp1ilem 9181 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∖ {𝑥}) = {𝑦, 𝑧} → 𝐴 = {𝑥, 𝑦, 𝑧})) |
7 | 6 | 2eximdv 1923 | . 2 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦∃𝑧(𝐴 ∖ {𝑥}) = {𝑦, 𝑧} → ∃𝑦∃𝑧 𝐴 = {𝑥, 𝑦, 𝑧})) |
8 | 2, 3, 4, 7 | enp1i 9182 | 1 ⊢ (𝐴 ≈ 3o → ∃𝑥∃𝑦∃𝑧 𝐴 = {𝑥, 𝑦, 𝑧}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∖ cdif 3906 {csn 4585 {cpr 4587 {ctp 4589 class class class wbr 5104 2oc2o 8399 3oc3o 8400 ≈ cen 8839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-ord 6319 df-on 6320 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-1o 8405 df-2o 8406 df-3o 8407 df-en 8843 |
This theorem is referenced by: en4 9186 hash3tr 14343 |
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