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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 2758 | . . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
2 | eueq 2923 | . . . 4 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴) | |
3 | 1, 2 | bitr3i 186 | . . 3 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃!𝑥 𝑥 = 𝐴) |
4 | 3 | biimpi 120 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃!𝑥 𝑥 = 𝐴) |
5 | df-mo 2042 | . 2 ⊢ (∃*𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 → ∃!𝑥 𝑥 = 𝐴)) | |
6 | 4, 5 | mpbir 146 | 1 ⊢ ∃*𝑥 𝑥 = 𝐴 |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∃wex 1503 ∃!weu 2038 ∃*wmo 2039 ∈ wcel 2160 Vcvv 2752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-v 2754 |
This theorem is referenced by: euxfr2dc 2937 reueq 2951 mosn 3643 sndisj 4014 disjxsn 4016 reusv1 4476 funopabeq 5271 funcnvsn 5280 fvmptg 5613 fvopab6 5633 ovmpt4g 6019 ovi3 6033 ov6g 6034 oprabex3 6154 1stconst 6246 2ndconst 6247 axaddf 7897 axmulf 7898 |
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