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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 528 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃*𝑥 𝑥 = 𝐴)) |
3 | isset 3485 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
4 | df-eu 2561 | . 2 ⊢ (∃!𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃*𝑥 𝑥 = 𝐴)) | |
5 | 2, 3, 4 | 3bitr4i 302 | 1 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1539 ∃wex 1779 ∈ wcel 2104 ∃*wmo 2530 ∃!weu 2560 Vcvv 3472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-ex 1780 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-v 3474 |
This theorem is referenced by: eueqi 3706 reuhypd 5418 mptfnf 6686 mptfng 6690 upxp 23349 iotasbc 43482 sprval 46447 |
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