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Mirrors > Home > MPE Home > Th. List > en4 | Structured version Visualization version GIF version |
Description: A set equinumerous to ordinal 4 is a quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Ref | Expression |
---|---|
en4 | ⊢ (𝐴 ≈ 4𝑜 → ∃𝑥∃𝑦∃𝑧∃𝑤 𝐴 = ({𝑥, 𝑦} ∪ {𝑧, 𝑤})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3onn 7608 | . 2 ⊢ 3𝑜 ∈ ω | |
2 | df-4o 7450 | . 2 ⊢ 4𝑜 = suc 3𝑜 | |
3 | en3 8082 | . 2 ⊢ ((𝐴 ∖ {𝑥}) ≈ 3𝑜 → ∃𝑦∃𝑧∃𝑤(𝐴 ∖ {𝑥}) = {𝑦, 𝑧, 𝑤}) | |
4 | qdassr 4233 | . . . . 5 ⊢ ({𝑥, 𝑦} ∪ {𝑧, 𝑤}) = ({𝑥} ∪ {𝑦, 𝑧, 𝑤}) | |
5 | 4 | enp1ilem 8079 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∖ {𝑥}) = {𝑦, 𝑧, 𝑤} → 𝐴 = ({𝑥, 𝑦} ∪ {𝑧, 𝑤}))) |
6 | 5 | eximdv 1833 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∃𝑤(𝐴 ∖ {𝑥}) = {𝑦, 𝑧, 𝑤} → ∃𝑤 𝐴 = ({𝑥, 𝑦} ∪ {𝑧, 𝑤}))) |
7 | 6 | 2eximdv 1835 | . 2 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦∃𝑧∃𝑤(𝐴 ∖ {𝑥}) = {𝑦, 𝑧, 𝑤} → ∃𝑦∃𝑧∃𝑤 𝐴 = ({𝑥, 𝑦} ∪ {𝑧, 𝑤}))) |
8 | 1, 2, 3, 7 | enp1i 8080 | 1 ⊢ (𝐴 ≈ 4𝑜 → ∃𝑥∃𝑦∃𝑧∃𝑤 𝐴 = ({𝑥, 𝑦} ∪ {𝑧, 𝑤})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∖ cdif 3537 ∪ cun 3538 {csn 4125 {cpr 4127 {ctp 4129 class class class wbr 4583 3𝑜c3o 7442 4𝑜c4o 7443 ≈ cen 7838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-1o 7447 df-2o 7448 df-3o 7449 df-4o 7450 df-er 7629 df-en 7842 df-fin 7845 |
This theorem is referenced by: (None) |
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