Theorem elisset 2817

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 2814	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		isset 2809	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylib 122	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   = wceq 1397  ∃wex 1540   ∈ wcel 2202  Vcvv 2802

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-v 2804

This theorem is referenced by:  elex22  2818  elex2  2819  ceqsalt  2829  ceqsalg  2831  cgsexg  2838  cgsex2g  2839  cgsex4g  2840  vtoclgft  2854  vtocleg  2877  vtoclegft  2878  spc2egv  2896  spc2gv  2897  spc3egv  2898  spc3gv  2899  eqvincg  2930  tpid3g  3787  iinexgm  4244  copsex2t  4337  copsex2g  4338  ralxfr2d  4561  rexxfr2d  4562  fliftf  5939  eloprabga  6107  ovmpt4g  6143  spc2ed  6397  eroveu  6794  supelti  7200  genpassl  7743  genpassu  7744  eqord1  8662  nn1suc  9161  bj-inex  16502