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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 | 2onn 8268 | . 2 ⊢ 2o ∈ ω | |
2 | df-3o 8106 | . 2 ⊢ 3o = suc 2o | |
3 | en2 8756 | . 2 ⊢ ((𝐴 ∖ {𝑥}) ≈ 2o → ∃𝑦∃𝑧(𝐴 ∖ {𝑥}) = {𝑦, 𝑧}) | |
4 | tpass 4690 | . . . 4 ⊢ {𝑥, 𝑦, 𝑧} = ({𝑥} ∪ {𝑦, 𝑧}) | |
5 | 4 | enp1ilem 8754 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∖ {𝑥}) = {𝑦, 𝑧} → 𝐴 = {𝑥, 𝑦, 𝑧})) |
6 | 5 | 2eximdv 1920 | . 2 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦∃𝑧(𝐴 ∖ {𝑥}) = {𝑦, 𝑧} → ∃𝑦∃𝑧 𝐴 = {𝑥, 𝑦, 𝑧})) |
7 | 1, 2, 3, 6 | enp1i 8755 | 1 ⊢ (𝐴 ≈ 3o → ∃𝑥∃𝑦∃𝑧 𝐴 = {𝑥, 𝑦, 𝑧}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∖ cdif 3935 {csn 4569 {cpr 4571 {ctp 4573 class class class wbr 5068 2oc2o 8098 3oc3o 8099 ≈ cen 8508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-1o 8104 df-2o 8105 df-3o 8106 df-er 8291 df-en 8512 df-fin 8515 |
This theorem is referenced by: en4 8758 hash3tr 13851 |
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