|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 3232 | . 2 ⊢ (𝐵 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐵) | |
| 2 | moeq 3712 | . . . 4 ⊢ ∃*𝑥 𝑥 = 𝐵 | |
| 3 | mormo 3384 | . . . 4 ⊢ (∃*𝑥 𝑥 = 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵) | |
| 4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵 | 
| 5 | reu5 3381 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝑥 = 𝐵 ↔ (∃𝑥 ∈ 𝐴 𝑥 = 𝐵 ∧ ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵)) | |
| 6 | 4, 5 | mpbiran2 710 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐵) | 
| 7 | 1, 6 | bitr4i 278 | 1 ⊢ (𝐵 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 𝑥 = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1539 ∈ wcel 2107 ∃*wmo 2537 ∃wrex 3069 ∃!wreu 3377 ∃*wrmo 3378 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-mo 2539 df-eu 2568 df-cleq 2728 df-clel 2815 df-rex 3070 df-rmo 3379 df-reu 3380 | 
| This theorem is referenced by: icoshftf1o 13515 addsq2reu 27485 euoreqb 47126 isuspgrimlem 47879 inlinecirc02preu 48714 | 
| Copyright terms: Public domain | W3C validator |