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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 | 1onn 8265 | . 2 ⊢ 1o ∈ ω | |
2 | df-2o 8103 | . 2 ⊢ 2o = suc 1o | |
3 | en1 8576 | . . 3 ⊢ ((𝐴 ∖ {𝑥}) ≈ 1o ↔ ∃𝑦(𝐴 ∖ {𝑥}) = {𝑦}) | |
4 | 3 | biimpi 218 | . 2 ⊢ ((𝐴 ∖ {𝑥}) ≈ 1o → ∃𝑦(𝐴 ∖ {𝑥}) = {𝑦}) |
5 | df-pr 4570 | . . . 4 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦}) | |
6 | 5 | enp1ilem 8752 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∖ {𝑥}) = {𝑦} → 𝐴 = {𝑥, 𝑦})) |
7 | 6 | eximdv 1918 | . 2 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦(𝐴 ∖ {𝑥}) = {𝑦} → ∃𝑦 𝐴 = {𝑥, 𝑦})) |
8 | 1, 2, 4, 7 | enp1i 8753 | 1 ⊢ (𝐴 ≈ 2o → ∃𝑥∃𝑦 𝐴 = {𝑥, 𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∖ cdif 3933 {csn 4567 {cpr 4569 class class class wbr 5066 1oc1o 8095 2oc2o 8096 ≈ cen 8506 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-om 7581 df-1o 8102 df-2o 8103 df-er 8289 df-en 8510 df-fin 8513 |
This theorem is referenced by: en3 8755 hash2pr 13828 pmtrrn2 18588 trsp2cyc 30765 en2pr 39926 pr2cv 39927 |
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