Theorem elisset 2830

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 2827	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		isset 2822	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylib 122	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   = wceq 1398  ∃wex 1541   ∈ wcel 2205  Vcvv 2815

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-v 2817

This theorem is referenced by:  elex22  2831  elex2  2832  ceqsalt  2842  ceqsalg  2844  cgsexg  2851  cgsex2g  2852  cgsex4g  2853  vtoclgft  2867  vtocleg  2890  vtoclegft  2891  spc2egv  2909  spc2gv  2910  spc3egv  2911  spc3gv  2912  eqvincg  2944  tpid3g  3812  iinexgm  4271  copsex2t  4366  copsex2g  4367  ralxfr2d  4590  rexxfr2d  4591  fliftf  5978  eloprabga  6148  ovmpt4g  6184  spc2ed  6442  eroveu  6873  supelti  7306  genpassl  7855  genpassu  7856  eqord1  8774  nn1suc  9273  bj-inex  16803