Theorem elisset 2766

Description: An element of a class exists. (Contributed by NM, 1-May-1995.)

Assertion

Ref	Expression
elisset	⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elisset

Step	Hyp	Ref	Expression
1		elex 2763	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		isset 2758	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylib 122	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   = wceq 1364  ∃wex 1503   ∈ 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-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-v 2754

This theorem is referenced by:  elex22  2767  elex2  2768  ceqsalt  2778  ceqsalg  2780  cgsexg  2787  cgsex2g  2788  cgsex4g  2789  vtoclgft  2802  vtocleg  2823  vtoclegft  2824  spc2egv  2842  spc2gv  2843  spc3egv  2844  spc3gv  2845  eqvincg  2876  tpid3g  3722  iinexgm  4172  copsex2t  4263  copsex2g  4264  ralxfr2d  4482  rexxfr2d  4483  fliftf  5821  eloprabga  5983  ovmpt4g  6019  spc2ed  6258  eroveu  6652  supelti  7031  genpassl  7553  genpassu  7554  eqord1  8470  nn1suc  8968  bj-inex  15117