Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 2099 | . . . 4 ⊢ ((𝐴 = 〈𝐵, 𝑥〉 ∧ 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) → 〈𝐵, 𝑥〉 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) | |
2 | moop2.1 | . . . . . 6 ⊢ 𝐵 ∈ V | |
3 | vex 2604 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | opth 3992 | . . . . 5 ⊢ (〈𝐵, 𝑥〉 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉 ↔ (𝐵 = ⦋𝑦 / 𝑥⦌𝐵 ∧ 𝑥 = 𝑦)) |
5 | 4 | simprbi 269 | . . . 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 1379 | . 2 ⊢ ∀𝑥∀𝑦((𝐴 = 〈𝐵, 𝑥〉 ∧ 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) → 𝑥 = 𝑦) |
8 | nfcsb1v 2938 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
9 | nfcv 2219 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
10 | 8, 9 | nfop 3586 | . . . 4 ⊢ Ⅎ𝑥〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉 |
11 | 10 | nfeq2 2230 | . . 3 ⊢ Ⅎ𝑥 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉 |
12 | csbeq1a 2916 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
13 | id 19 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
14 | 12, 13 | opeq12d 3578 | . . . 4 ⊢ (𝑥 = 𝑦 → 〈𝐵, 𝑥〉 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) |
15 | 14 | eqeq2d 2092 | . . 3 ⊢ (𝑥 = 𝑦 → (𝐴 = 〈𝐵, 𝑥〉 ↔ 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉)) |
16 | 11, 15 | mo4f 2001 | . 2 ⊢ (∃*𝑥 𝐴 = 〈𝐵, 𝑥〉 ↔ ∀𝑥∀𝑦((𝐴 = 〈𝐵, 𝑥〉 ∧ 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) → 𝑥 = 𝑦)) |
17 | 7, 16 | mpbir 144 | 1 ⊢ ∃*𝑥 𝐴 = 〈𝐵, 𝑥〉 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∀wal 1282 = wceq 1284 ∈ wcel 1433 ∃*wmo 1942 Vcvv 2601 ⦋csb 2908 〈cop 3401 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |