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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 3267 | . 2 ⊢ (𝐵 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐵) | |
2 | moeq 3697 | . . . 4 ⊢ ∃*𝑥 𝑥 = 𝐵 | |
3 | mormo 3429 | . . . 4 ⊢ (∃*𝑥 𝑥 = 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ∃*𝑥 ∈ 𝐴 𝑥 = 𝐵 |
5 | reu5 3430 | . . 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 280 | 1 ⊢ (𝐵 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 𝑥 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ∈ wcel 2110 ∃*wmo 2616 ∃wrex 3139 ∃!wreu 3140 ∃*wrmo 3141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-mo 2618 df-eu 2650 df-cleq 2814 df-clel 2893 df-rex 3144 df-reu 3145 df-rmo 3146 |
This theorem is referenced by: icoshftf1o 12854 addsq2reu 26010 euoreqb 43302 inlinecirc02preu 44769 |
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