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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 2801 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 = 〈𝑎, 𝑏〉 ↔ 〈𝑎, 𝑏〉 = 〈𝑥, 𝑦〉) | |
2 | vex 3439 | . . . . . . . . 9 ⊢ 𝑎 ∈ V | |
3 | vex 3439 | . . . . . . . . 9 ⊢ 𝑏 ∈ V | |
4 | 2, 3 | opth 5263 | . . . . . . . 8 ⊢ (〈𝑎, 𝑏〉 = 〈𝑥, 𝑦〉 ↔ (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)) |
5 | 1, 4 | bitri 276 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 = 〈𝑎, 𝑏〉 ↔ (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)) |
6 | 5 | imbi2i 337 | . . . . . 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 3164 | . . . 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 3260 | . . 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 6021 | . 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 4382 | . 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 304 | 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 207 ∧ wa 396 = wceq 1522 ∈ wcel 2080 ∀wral 3104 ∃wrex 3105 ∃!wreu 3106 〈cop 4480 × cxp 5444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pr 5224 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-nul 4214 df-if 4384 df-sn 4475 df-pr 4477 df-op 4481 df-iun 4829 df-opab 5027 df-xp 5452 df-rel 5453 |
This theorem is referenced by: opreu2reu 6024 2sqreuop 25720 2sqreuopnn 25721 2sqreuoplt 25722 2sqreuopltb 25723 2sqreuopnnlt 25724 2sqreuopnnltb 25725 opreu2reu1 29931 |
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