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Theorem elisset 2751
Description: An element of a class exists. (Contributed by NM, 1-May-1995.)
Assertion
Ref Expression
elisset (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elisset
StepHypRef Expression
1 elex 2748 . 2 (𝐴𝑉𝐴 ∈ V)
2 isset 2743 . 2 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
31, 2sylib 122 1 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wex 1492  wcel 2148  Vcvv 2737
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-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2739
This theorem is referenced by:  elex22  2752  elex2  2753  ceqsalt  2763  ceqsalg  2765  cgsexg  2772  cgsex2g  2773  cgsex4g  2774  vtoclgft  2787  vtocleg  2808  vtoclegft  2809  spc2egv  2827  spc2gv  2828  spc3egv  2829  spc3gv  2830  eqvincg  2861  tpid3g  3707  iinexgm  4154  copsex2t  4245  copsex2g  4246  ralxfr2d  4464  rexxfr2d  4465  fliftf  5799  eloprabga  5961  ovmpt4g  5996  spc2ed  6233  eroveu  6625  supelti  7000  genpassl  7522  genpassu  7523  eqord1  8439  nn1suc  8937  bj-inex  14629
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