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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 9278 | . 2 ⊢ ({𝐴, 𝐵} ≈ 2o → ∃𝑥∃𝑦{𝐴, 𝐵} = {𝑥, 𝑦}) | |
2 | breq1 5142 | . . . 4 ⊢ ({𝐴, 𝐵} = {𝑥, 𝑦} → ({𝐴, 𝐵} ≈ 2o ↔ {𝑥, 𝑦} ≈ 2o)) | |
3 | vex 3470 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
4 | vex 3470 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
5 | pr2ne 9996 | . . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ({𝑥, 𝑦} ≈ 2o ↔ 𝑥 ≠ 𝑦)) | |
6 | 5 | el2v 3474 | . . . . . . . 8 ⊢ ({𝑥, 𝑦} ≈ 2o ↔ 𝑥 ≠ 𝑦) |
7 | 6 | biimpi 215 | . . . . . . 7 ⊢ ({𝑥, 𝑦} ≈ 2o → 𝑥 ≠ 𝑦) |
8 | preq12nebg 4856 | . . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑥 ≠ 𝑦) → ({𝑥, 𝑦} = {𝐴, 𝐵} ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∨ (𝑥 = 𝐵 ∧ 𝑦 = 𝐴)))) | |
9 | eqvisset 3484 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V) | |
10 | eqvisset 3484 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → 𝐵 ∈ V) | |
11 | 9, 10 | anim12i 612 | . . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
12 | eqvisset 3484 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → 𝐵 ∈ V) | |
13 | eqvisset 3484 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → 𝐴 ∈ V) | |
14 | 12, 13 | anim12ci 613 | . . . . . . . . 9 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
15 | 11, 14 | jaoi 854 | . . . . . . . 8 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∨ (𝑥 = 𝐵 ∧ 𝑦 = 𝐴)) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
16 | 8, 15 | syl6bi 253 | . . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑥 ≠ 𝑦) → ({𝑥, 𝑦} = {𝐴, 𝐵} → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
17 | 3, 4, 7, 16 | mp3an12i 1461 | . . . . . 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 2732 | . . . 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 1927 | . 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 395 ∨ wo 844 ∧ w3a 1084 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2932 Vcvv 3466 {cpr 4623 class class class wbr 5139 2oc2o 8456 ≈ cen 8933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-ord 6358 df-on 6359 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-1o 8462 df-2o 8463 df-en 8937 |
This theorem is referenced by: pr2el1 42814 pr2cv1 42815 pr2el2 42816 pr2cv2 42817 pren2 42818 |
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