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Mirrors > Home > NFE Home > Th. List > eueq1 | GIF version |
Description: Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
Ref | Expression |
---|---|
eueq1.1 | ⊢ A ∈ V |
Ref | Expression |
---|---|
eueq1 | ⊢ ∃!x x = A |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eueq1.1 | . 2 ⊢ A ∈ V | |
2 | eueq 3008 | . 2 ⊢ (A ∈ V ↔ ∃!x x = A) | |
3 | 1, 2 | mpbi 199 | 1 ⊢ ∃!x x = A |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∈ wcel 1710 ∃!weu 2204 Vcvv 2859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-v 2861 |
This theorem is referenced by: eueq2 3010 eueq3 3011 fnopab2 5208 fsn 5432 composefn 5818 |
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