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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 3229 | . 2 ⊢ (𝐵 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐵) | |
2 | moeq 3699 | . . . 4 ⊢ ∃*𝑥 𝑥 = 𝐵 | |
3 | mormo 3380 | . . . 4 ⊢ (∃*𝑥 𝑥 = 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵 |
5 | reu5 3377 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝑥 = 𝐵 ↔ (∃𝑥 ∈ 𝐴 𝑥 = 𝐵 ∧ ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵)) | |
6 | 4, 5 | mpbiran2 708 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐵) |
7 | 1, 6 | bitr4i 277 | 1 ⊢ (𝐵 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 𝑥 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 ∃*wmo 2531 ∃wrex 3069 ∃!wreu 3373 ∃*wrmo 3374 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-mo 2533 df-eu 2562 df-cleq 2723 df-clel 2809 df-rex 3070 df-rmo 3375 df-reu 3376 |
This theorem is referenced by: icoshftf1o 13433 addsq2reu 26870 euoreqb 45589 inlinecirc02preu 47122 |
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