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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 3606 and eueq 3607. (Proof shortened by BJ, 24-Sep-2022.) |
Ref | Expression |
---|---|
eueq | ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moeq 3606 | . . 3 ⊢ ∃*𝑥 𝑥 = 𝐴 | |
2 | 1 | biantru 533 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃*𝑥 𝑥 = 𝐴)) |
3 | isset 3411 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
4 | df-eu 2570 | . 2 ⊢ (∃!𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃*𝑥 𝑥 = 𝐴)) | |
5 | 2, 3, 4 | 3bitr4i 306 | 1 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1542 ∃wex 1786 ∈ wcel 2114 ∃*wmo 2538 ∃!weu 2569 Vcvv 3398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-v 3400 |
This theorem is referenced by: eueqi 3608 reuhypd 5286 mptfnf 6472 mptfng 6476 upxp 22376 iotasbc 41597 sprval 44494 |
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