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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 2605 | . . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
2 | eueq 2763 | . . . 4 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴) | |
3 | 1, 2 | bitr3i 184 | . . 3 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃!𝑥 𝑥 = 𝐴) |
4 | 3 | biimpi 118 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃!𝑥 𝑥 = 𝐴) |
5 | df-mo 1945 | . 2 ⊢ (∃*𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 → ∃!𝑥 𝑥 = 𝐴)) | |
6 | 4, 5 | mpbir 144 | 1 ⊢ ∃*𝑥 𝑥 = 𝐴 |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ∃wex 1421 ∈ wcel 1433 ∃!weu 1941 ∃*wmo 1942 Vcvv 2601 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-v 2603 |
This theorem is referenced by: euxfr2dc 2777 reueq 2789 mosn 3429 sndisj 3781 disjxsn 3783 reusv1 4208 funopabeq 4956 funcnvsn 4965 fvmptg 5269 fvopab6 5285 ovmpt4g 5643 ovi3 5657 ov6g 5658 oprabex3 5776 1stconst 5862 2ndconst 5863 |
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