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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 2819 | . . . 4 ⊢ ((𝐴 = 〈𝐵, 𝑥〉 ∧ 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) → 〈𝐵, 𝑥〉 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) | |
2 | moop2.1 | . . . . . 6 ⊢ 𝐵 ∈ V | |
3 | vex 3444 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | opth 5333 | . . . . 5 ⊢ (〈𝐵, 𝑥〉 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉 ↔ (𝐵 = ⦋𝑦 / 𝑥⦌𝐵 ∧ 𝑥 = 𝑦)) |
5 | 4 | simprbi 500 | . . . 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 1798 | . 2 ⊢ ∀𝑥∀𝑦((𝐴 = 〈𝐵, 𝑥〉 ∧ 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) → 𝑥 = 𝑦) |
8 | nfcsb1v 3852 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
9 | nfcv 2955 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
10 | 8, 9 | nfop 4781 | . . . 4 ⊢ Ⅎ𝑥〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉 |
11 | 10 | nfeq2 2972 | . . 3 ⊢ Ⅎ𝑥 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉 |
12 | csbeq1a 3842 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
13 | id 22 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
14 | 12, 13 | opeq12d 4773 | . . . 4 ⊢ (𝑥 = 𝑦 → 〈𝐵, 𝑥〉 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) |
15 | 14 | eqeq2d 2809 | . . 3 ⊢ (𝑥 = 𝑦 → (𝐴 = 〈𝐵, 𝑥〉 ↔ 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉)) |
16 | 11, 15 | mo4f 2626 | . 2 ⊢ (∃*𝑥 𝐴 = 〈𝐵, 𝑥〉 ↔ ∀𝑥∀𝑦((𝐴 = 〈𝐵, 𝑥〉 ∧ 𝐴 = 〈⦋𝑦 / 𝑥⦌𝐵, 𝑦〉) → 𝑥 = 𝑦)) |
17 | 7, 16 | mpbir 234 | 1 ⊢ ∃*𝑥 𝐴 = 〈𝐵, 𝑥〉 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1536 = wceq 1538 ∈ wcel 2111 ∃*wmo 2596 Vcvv 3441 ⦋csb 3828 〈cop 4531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 |
This theorem is referenced by: euop2 5367 |
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