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Mirrors > Home > MPE Home > Th. List > Mathboxes > pr2cv | Structured version Visualization version GIF version |
Description: If an unordered pair is equinumerous to ordinal two, then both parts are sets. (Contributed by RP, 8-Oct-2023.) |
Ref | Expression |
---|---|
pr2cv | ⊢ ({𝐴, 𝐵} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en2 9307 | . 2 ⊢ ({𝐴, 𝐵} ≈ 2o → ∃𝑥∃𝑦{𝐴, 𝐵} = {𝑥, 𝑦}) | |
2 | breq1 5152 | . . . 4 ⊢ ({𝐴, 𝐵} = {𝑥, 𝑦} → ({𝐴, 𝐵} ≈ 2o ↔ {𝑥, 𝑦} ≈ 2o)) | |
3 | vex 3481 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
4 | vex 3481 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
5 | pr2ne 10035 | . . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ({𝑥, 𝑦} ≈ 2o ↔ 𝑥 ≠ 𝑦)) | |
6 | 5 | el2v 3484 | . . . . . . . 8 ⊢ ({𝑥, 𝑦} ≈ 2o ↔ 𝑥 ≠ 𝑦) |
7 | 6 | biimpi 216 | . . . . . . 7 ⊢ ({𝑥, 𝑦} ≈ 2o → 𝑥 ≠ 𝑦) |
8 | preq12nebg 4870 | . . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑥 ≠ 𝑦) → ({𝑥, 𝑦} = {𝐴, 𝐵} ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∨ (𝑥 = 𝐵 ∧ 𝑦 = 𝐴)))) | |
9 | eqvisset 3497 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V) | |
10 | eqvisset 3497 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → 𝐵 ∈ V) | |
11 | 9, 10 | anim12i 612 | . . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
12 | eqvisset 3497 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → 𝐵 ∈ V) | |
13 | eqvisset 3497 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → 𝐴 ∈ V) | |
14 | 12, 13 | anim12ci 613 | . . . . . . . . 9 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
15 | 11, 14 | jaoi 856 | . . . . . . . 8 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∨ (𝑥 = 𝐵 ∧ 𝑦 = 𝐴)) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
16 | 8, 15 | biimtrdi 253 | . . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑥 ≠ 𝑦) → ({𝑥, 𝑦} = {𝐴, 𝐵} → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
17 | 3, 4, 7, 16 | mp3an12i 1463 | . . . . . 6 ⊢ ({𝑥, 𝑦} ≈ 2o → ({𝑥, 𝑦} = {𝐴, 𝐵} → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
18 | 17 | com12 32 | . . . . 5 ⊢ ({𝑥, 𝑦} = {𝐴, 𝐵} → ({𝑥, 𝑦} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
19 | 18 | eqcoms 2741 | . . . 4 ⊢ ({𝐴, 𝐵} = {𝑥, 𝑦} → ({𝑥, 𝑦} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
20 | 2, 19 | sylbid 240 | . . 3 ⊢ ({𝐴, 𝐵} = {𝑥, 𝑦} → ({𝐴, 𝐵} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
21 | 20 | exlimivv 1928 | . 2 ⊢ (∃𝑥∃𝑦{𝐴, 𝐵} = {𝑥, 𝑦} → ({𝐴, 𝐵} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
22 | 1, 21 | mpcom 38 | 1 ⊢ ({𝐴, 𝐵} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 846 ∧ w3a 1085 = wceq 1535 ∃wex 1774 ∈ wcel 2104 ≠ wne 2936 Vcvv 3477 {cpr 4632 class class class wbr 5149 2oc2o 8493 ≈ cen 8975 |
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 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3377 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-opab 5212 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-ord 6383 df-on 6384 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-1o 8499 df-2o 8500 df-en 8979 |
This theorem is referenced by: pr2el1 43497 pr2cv1 43498 pr2el2 43499 pr2cv2 43500 pren2 43501 |
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