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Mirrors > Home > ILE Home > Th. List > reueq | GIF version |
Description: Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.) |
Ref | Expression |
---|---|
reueq | ⊢ (𝐵 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 𝑥 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | risset 2505 | . 2 ⊢ (𝐵 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐵) | |
2 | moeq 2914 | . . . 4 ⊢ ∃*𝑥 𝑥 = 𝐵 | |
3 | mormo 2689 | . . . 4 ⊢ (∃*𝑥 𝑥 = 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵 |
5 | reu5 2690 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝑥 = 𝐵 ↔ (∃𝑥 ∈ 𝐴 𝑥 = 𝐵 ∧ ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵)) | |
6 | 4, 5 | mpbiran2 941 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐵) |
7 | 1, 6 | bitr4i 187 | 1 ⊢ (𝐵 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 𝑥 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1353 ∃*wmo 2027 ∈ wcel 2148 ∃wrex 2456 ∃!wreu 2457 ∃*wrmo 2458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-rex 2461 df-reu 2462 df-rmo 2463 df-v 2741 |
This theorem is referenced by: divfnzn 9623 icoshftf1o 9993 |
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