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Mirrors > Home > MPE Home > Th. List > reueq | Structured version Visualization version 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 3044 | . 2 ⊢ (𝐵 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐵) | |
2 | moeq 3349 | . . . 4 ⊢ ∃*𝑥 𝑥 = 𝐵 | |
3 | mormo 3135 | . . . 4 ⊢ (∃*𝑥 𝑥 = 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵 |
5 | reu5 3136 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝑥 = 𝐵 ↔ (∃𝑥 ∈ 𝐴 𝑥 = 𝐵 ∧ ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵)) | |
6 | 4, 5 | mpbiran2 956 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐵) |
7 | 1, 6 | bitr4i 266 | 1 ⊢ (𝐵 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 𝑥 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∃*wmo 2459 ∃wrex 2897 ∃!wreu 2898 ∃*wrmo 2899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-rex 2902 df-reu 2903 df-rmo 2904 df-v 3175 |
This theorem is referenced by: icoshftf1o 12166 |
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