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Mirrors > Home > MPE Home > Th. List > opreu2reurex | Structured version Visualization version GIF version |
Description: There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 24-Jun-2023.) (Revised by AV, 1-Jul-2023.) |
Ref | Expression |
---|---|
opreu2reurex.a | ⊢ (𝑝 = 〈𝑎, 𝑏〉 → (𝜑 ↔ 𝜒)) |
Ref | Expression |
---|---|
opreu2reurex | ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2738 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 = 〈𝑎, 𝑏〉 ↔ 〈𝑎, 𝑏〉 = 〈𝑥, 𝑦〉) | |
2 | vex 3477 | . . . . . . . . 9 ⊢ 𝑎 ∈ V | |
3 | vex 3477 | . . . . . . . . 9 ⊢ 𝑏 ∈ V | |
4 | 2, 3 | opth 5476 | . . . . . . . 8 ⊢ (〈𝑎, 𝑏〉 = 〈𝑥, 𝑦〉 ↔ (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)) |
5 | 1, 4 | bitri 275 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 = 〈𝑎, 𝑏〉 ↔ (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)) |
6 | 5 | imbi2i 336 | . . . . . 6 ⊢ ((𝜒 → 〈𝑥, 𝑦〉 = 〈𝑎, 𝑏〉) ↔ (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) |
7 | 6 | a1i 11 | . . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → ((𝜒 → 〈𝑥, 𝑦〉 = 〈𝑎, 𝑏〉) ↔ (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)))) |
8 | 7 | 2ralbidva 3215 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → 〈𝑥, 𝑦〉 = 〈𝑎, 𝑏〉) ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)))) |
9 | 8 | 2rexbiia 3214 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → 〈𝑥, 𝑦〉 = 〈𝑎, 𝑏〉) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) |
10 | 9 | anbi2i 622 | . 2 ⊢ ((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → 〈𝑥, 𝑦〉 = 〈𝑎, 𝑏〉)) ↔ (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)))) |
11 | opreu2reurex.a | . . 3 ⊢ (𝑝 = 〈𝑎, 𝑏〉 → (𝜑 ↔ 𝜒)) | |
12 | 11 | reu3op 6291 | . 2 ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → 〈𝑥, 𝑦〉 = 〈𝑎, 𝑏〉))) |
13 | 2reu4 4526 | . 2 ⊢ ((∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒) ↔ (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)))) | |
14 | 10, 12, 13 | 3bitr4i 303 | 1 ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ∃wrex 3069 ∃!wreu 3373 〈cop 4634 × cxp 5674 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-iun 4999 df-opab 5211 df-xp 5682 df-rel 5683 |
This theorem is referenced by: opreu2reu 6294 2sqreuop 27307 2sqreuopnn 27308 2sqreuoplt 27309 2sqreuopltb 27310 2sqreuopnnlt 27311 2sqreuopnnltb 27312 opreu2reu1 32156 |
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