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Theorem moeq 2905
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 2736 . . . 4 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2 eueq 2901 . . . 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 2023 . 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 1348  wex 1485  ∃!weu 2019  ∃*wmo 2020  wcel 2141  Vcvv 2730
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732
This theorem is referenced by:  euxfr2dc  2915  reueq  2929  mosn  3619  sndisj  3985  disjxsn  3987  reusv1  4443  funopabeq  5234  funcnvsn  5243  fvmptg  5572  fvopab6  5592  ovmpt4g  5975  ovi3  5989  ov6g  5990  oprabex3  6108  1stconst  6200  2ndconst  6201  axaddf  7830  axmulf  7831
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