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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 9277 | . 2 ⊢ ({𝐴, 𝐵} ≈ 2o → ∃𝑥∃𝑦{𝐴, 𝐵} = {𝑥, 𝑦}) | |
2 | breq1 5150 | . . . 4 ⊢ ({𝐴, 𝐵} = {𝑥, 𝑦} → ({𝐴, 𝐵} ≈ 2o ↔ {𝑥, 𝑦} ≈ 2o)) | |
3 | vex 3478 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
4 | vex 3478 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
5 | pr2ne 9995 | . . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ({𝑥, 𝑦} ≈ 2o ↔ 𝑥 ≠ 𝑦)) | |
6 | 5 | el2v 3482 | . . . . . . . 8 ⊢ ({𝑥, 𝑦} ≈ 2o ↔ 𝑥 ≠ 𝑦) |
7 | 6 | biimpi 215 | . . . . . . 7 ⊢ ({𝑥, 𝑦} ≈ 2o → 𝑥 ≠ 𝑦) |
8 | preq12nebg 4862 | . . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑥 ≠ 𝑦) → ({𝑥, 𝑦} = {𝐴, 𝐵} ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∨ (𝑥 = 𝐵 ∧ 𝑦 = 𝐴)))) | |
9 | eqvisset 3491 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V) | |
10 | eqvisset 3491 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → 𝐵 ∈ V) | |
11 | 9, 10 | anim12i 613 | . . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
12 | eqvisset 3491 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → 𝐵 ∈ V) | |
13 | eqvisset 3491 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → 𝐴 ∈ V) | |
14 | 12, 13 | anim12ci 614 | . . . . . . . . 9 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
15 | 11, 14 | jaoi 855 | . . . . . . . 8 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∨ (𝑥 = 𝐵 ∧ 𝑦 = 𝐴)) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
16 | 8, 15 | syl6bi 252 | . . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑥 ≠ 𝑦) → ({𝑥, 𝑦} = {𝐴, 𝐵} → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
17 | 3, 4, 7, 16 | mp3an12i 1465 | . . . . . 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 2740 | . . . 4 ⊢ ({𝐴, 𝐵} = {𝑥, 𝑦} → ({𝑥, 𝑦} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
20 | 2, 19 | sylbid 239 | . . 3 ⊢ ({𝐴, 𝐵} = {𝑥, 𝑦} → ({𝐴, 𝐵} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
21 | 20 | exlimivv 1935 | . 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 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 Vcvv 3474 {cpr 4629 class class class wbr 5147 2oc2o 8456 ≈ cen 8932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6364 df-on 6365 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-1o 8462 df-2o 8463 df-en 8936 |
This theorem is referenced by: pr2el1 42285 pr2cv1 42286 pr2el2 42287 pr2cv2 42288 pren2 42289 |
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