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Theorem moeq 2927
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 2758 . . . 4 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2 eueq 2923 . . . 4 (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
31, 2bitr3i 186 . . 3 (∃𝑥 𝑥 = 𝐴 ↔ ∃!𝑥 𝑥 = 𝐴)
43biimpi 120 . 2 (∃𝑥 𝑥 = 𝐴 → ∃!𝑥 𝑥 = 𝐴)
5 df-mo 2042 . 2 (∃*𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 → ∃!𝑥 𝑥 = 𝐴))
64, 5mpbir 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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