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Theorem moeq 2859
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 2692 . . . 4 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2 eueq 2855 . . . 4 (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
31, 2bitr3i 185 . . 3 (∃𝑥 𝑥 = 𝐴 ↔ ∃!𝑥 𝑥 = 𝐴)
43biimpi 119 . 2 (∃𝑥 𝑥 = 𝐴 → ∃!𝑥 𝑥 = 𝐴)
5 df-mo 2003 . 2 (∃*𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 → ∃!𝑥 𝑥 = 𝐴))
64, 5mpbir 145 1 ∃*𝑥 𝑥 = 𝐴
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wex 1468  wcel 1480  ∃!weu 1999  ∃*wmo 2000  Vcvv 2686
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688
This theorem is referenced by:  euxfr2dc  2869  reueq  2883  mosn  3560  sndisj  3925  disjxsn  3927  reusv1  4379  funopabeq  5159  funcnvsn  5168  fvmptg  5497  fvopab6  5517  ovmpt4g  5893  ovi3  5907  ov6g  5908  oprabex3  6027  1stconst  6118  2ndconst  6119  axaddf  7683  axmulf  7684
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