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Mirrors > Home > MPE Home > Th. List > eueq | Structured version Visualization version GIF version |
Description: A class is a set if and only if there exists a unique set equal to it. (Contributed by NM, 25-Nov-1994.) Shorten combined proofs of moeq 3704 and eueq 3705. (Proof shortened by BJ, 24-Sep-2022.) |
Ref | Expression |
---|---|
eueq | ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moeq 3704 | . . 3 ⊢ ∃*𝑥 𝑥 = 𝐴 | |
2 | 1 | biantru 531 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃*𝑥 𝑥 = 𝐴)) |
3 | isset 3488 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
4 | df-eu 2564 | . 2 ⊢ (∃!𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃*𝑥 𝑥 = 𝐴)) | |
5 | 2, 3, 4 | 3bitr4i 303 | 1 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∃*wmo 2533 ∃!weu 2563 Vcvv 3475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 |
This theorem is referenced by: eueqi 3706 reuhypd 5418 mptfnf 6686 mptfng 6690 upxp 23127 iotasbc 43178 sprval 46147 |
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