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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 2625 | . . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
2 | eueq 2786 | . . . 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 1952 | . 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 1289 ∃wex 1426 ∈ wcel 1438 ∃!weu 1948 ∃*wmo 1949 Vcvv 2619 |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-v 2621 |
This theorem is referenced by: euxfr2dc 2800 reueq 2814 mosn 3479 sndisj 3841 disjxsn 3843 reusv1 4280 funopabeq 5050 funcnvsn 5059 fvmptg 5380 fvopab6 5396 ovmpt4g 5767 ovi3 5781 ov6g 5782 oprabex3 5900 1stconst 5986 2ndconst 5987 |
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