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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 2744 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 = 〈𝑎, 𝑏〉 ↔ 〈𝑎, 𝑏〉 = 〈𝑥, 𝑦〉) | |
| 2 | vex 3484 | . . . . . . . . 9 ⊢ 𝑎 ∈ V | |
| 3 | vex 3484 | . . . . . . . . 9 ⊢ 𝑏 ∈ V | |
| 4 | 2, 3 | opth 5481 | . . . . . . . 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 3219 | . . . 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 3218 | . . 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 623 | . 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 6312 | . 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 4523 | . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ∃!wreu 3378 〈cop 4632 × cxp 5683 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-iun 4993 df-opab 5206 df-xp 5691 df-rel 5692 |
| This theorem is referenced by: opreu2reu 6315 2sqreuop 27506 2sqreuopnn 27507 2sqreuoplt 27508 2sqreuopltb 27509 2sqreuopnnlt 27510 2sqreuopnnltb 27511 opreu2reu1 32503 |
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