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Mirrors > Home > ILE Home > Th. List > moeq | GIF version |
Description: There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.) |
Ref | Expression |
---|---|
moeq | ⊢ ∃*𝑥 𝑥 = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isset 2736 | . . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
2 | eueq 2901 | . . . 4 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴) | |
3 | 1, 2 | bitr3i 185 | . . 3 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃!𝑥 𝑥 = 𝐴) |
4 | 3 | biimpi 119 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃!𝑥 𝑥 = 𝐴) |
5 | df-mo 2023 | . 2 ⊢ (∃*𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 → ∃!𝑥 𝑥 = 𝐴)) | |
6 | 4, 5 | mpbir 145 | 1 ⊢ ∃*𝑥 𝑥 = 𝐴 |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∃wex 1485 ∃!weu 2019 ∃*wmo 2020 ∈ wcel 2141 Vcvv 2730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-v 2732 |
This theorem is referenced by: euxfr2dc 2915 reueq 2929 mosn 3619 sndisj 3985 disjxsn 3987 reusv1 4443 funopabeq 5234 funcnvsn 5243 fvmptg 5572 fvopab6 5592 ovmpt4g 5975 ovi3 5989 ov6g 5990 oprabex3 6108 1stconst 6200 2ndconst 6201 axaddf 7830 axmulf 7831 |
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