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| Mirrors > Home > MPE Home > Th. List > moop2 | Structured version Visualization version 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 2761 | . . . 4 ⊢ ((𝐴 = 〈𝐵, 𝑥〉 ∧ 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) → 〈𝐵, 𝑥〉 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) | |
| 2 | moop2.1 | . . . . . 6 ⊢ 𝐵 ∈ V | |
| 3 | vex 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 4 | 2, 3 | opth 5481 | . . . . 5 ⊢ (〈𝐵, 𝑥〉 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉 ↔ (𝐵 = ⦋𝑦 / 𝑥⦌𝐵 ∧ 𝑥 = 𝑦)) |
| 5 | 4 | simprbi 496 | . . . 4 ⊢ (〈𝐵, 𝑥〉 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉 → 𝑥 = 𝑦) |
| 6 | 1, 5 | syl 17 | . . 3 ⊢ ((𝐴 = 〈𝐵, 𝑥〉 ∧ 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) → 𝑥 = 𝑦) |
| 7 | 6 | gen2 1796 | . 2 ⊢ ∀𝑥∀𝑦((𝐴 = 〈𝐵, 𝑥〉 ∧ 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) → 𝑥 = 𝑦) |
| 8 | nfcsb1v 3923 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
| 9 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
| 10 | 8, 9 | nfop 4889 | . . . 4 ⊢ Ⅎ𝑥〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉 |
| 11 | 10 | nfeq2 2923 | . . 3 ⊢ Ⅎ𝑥 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉 |
| 12 | csbeq1a 3913 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
| 13 | id 22 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
| 14 | 12, 13 | opeq12d 4881 | . . . 4 ⊢ (𝑥 = 𝑦 → 〈𝐵, 𝑥〉 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) |
| 15 | 14 | eqeq2d 2748 | . . 3 ⊢ (𝑥 = 𝑦 → (𝐴 = 〈𝐵, 𝑥〉 ↔ 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉)) |
| 16 | 11, 15 | mo4f 2567 | . 2 ⊢ (∃*𝑥 𝐴 = 〈𝐵, 𝑥〉 ↔ ∀𝑥∀𝑦((𝐴 = 〈𝐵, 𝑥〉 ∧ 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) → 𝑥 = 𝑦)) |
| 17 | 7, 16 | mpbir 231 | 1 ⊢ ∃*𝑥 𝐴 = 〈𝐵, 𝑥〉 |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∃*wmo 2538 Vcvv 3480 ⦋csb 3899 〈cop 4632 |
| 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-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 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 |
| This theorem is referenced by: euop2 5517 |
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