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Mirrors > Home > ILE Home > Th. List > moop2 | GIF version |
Description: "At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
moop2.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
moop2 | ⊢ ∃*𝑥 𝐴 = 〈𝐵, 𝑥〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqtr2 2158 | . . . 4 ⊢ ((𝐴 = 〈𝐵, 𝑥〉 ∧ 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) → 〈𝐵, 𝑥〉 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) | |
2 | moop2.1 | . . . . . 6 ⊢ 𝐵 ∈ V | |
3 | vex 2689 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | opth 4159 | . . . . 5 ⊢ (〈𝐵, 𝑥〉 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉 ↔ (𝐵 = ⦋𝑦 / 𝑥⦌𝐵 ∧ 𝑥 = 𝑦)) |
5 | 4 | simprbi 273 | . . . 4 ⊢ (〈𝐵, 𝑥〉 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉 → 𝑥 = 𝑦) |
6 | 1, 5 | syl 14 | . . 3 ⊢ ((𝐴 = 〈𝐵, 𝑥〉 ∧ 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) → 𝑥 = 𝑦) |
7 | 6 | gen2 1426 | . 2 ⊢ ∀𝑥∀𝑦((𝐴 = 〈𝐵, 𝑥〉 ∧ 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) → 𝑥 = 𝑦) |
8 | nfcsb1v 3035 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
9 | nfcv 2281 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
10 | 8, 9 | nfop 3721 | . . . 4 ⊢ Ⅎ𝑥〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉 |
11 | 10 | nfeq2 2293 | . . 3 ⊢ Ⅎ𝑥 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉 |
12 | csbeq1a 3012 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
13 | id 19 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
14 | 12, 13 | opeq12d 3713 | . . . 4 ⊢ (𝑥 = 𝑦 → 〈𝐵, 𝑥〉 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) |
15 | 14 | eqeq2d 2151 | . . 3 ⊢ (𝑥 = 𝑦 → (𝐴 = 〈𝐵, 𝑥〉 ↔ 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉)) |
16 | 11, 15 | mo4f 2059 | . 2 ⊢ (∃*𝑥 𝐴 = 〈𝐵, 𝑥〉 ↔ ∀𝑥∀𝑦((𝐴 = 〈𝐵, 𝑥〉 ∧ 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) → 𝑥 = 𝑦)) |
17 | 7, 16 | mpbir 145 | 1 ⊢ ∃*𝑥 𝐴 = 〈𝐵, 𝑥〉 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1329 = wceq 1331 ∈ wcel 1480 ∃*wmo 2000 Vcvv 2686 ⦋csb 3003 〈cop 3530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 |
This theorem is referenced by: (None) |
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