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Theorem moeq 2739
Description: There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.)
Assertion
Ref Expression
moeq ∃*𝑥 𝑥 = 𝐴
Distinct variable group:   𝑥,𝐴

Proof of Theorem moeq
StepHypRef Expression
1 isset 2578 . . . 4 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2 eueq 2735 . . . 4 (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
31, 2bitr3i 179 . . 3 (∃𝑥 𝑥 = 𝐴 ↔ ∃!𝑥 𝑥 = 𝐴)
43biimpi 117 . 2 (∃𝑥 𝑥 = 𝐴 → ∃!𝑥 𝑥 = 𝐴)
5 df-mo 1920 . 2 (∃*𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 → ∃!𝑥 𝑥 = 𝐴))
64, 5mpbir 138 1 ∃*𝑥 𝑥 = 𝐴
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wex 1397  wcel 1409  ∃!weu 1916  ∃*wmo 1917  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by:  euxfr2dc  2749  reueq  2761  mosn  3434  sndisj  3788  disjxsn  3790  reusv1  4218  funopabeq  4964  funcnvsn  4973  fvmptg  5276  fvopab6  5292  ovmpt4g  5651  ovi3  5665  ov6g  5666  oprabex3  5784  1stconst  5870  2ndconst  5871
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