Theorem elisset 2815

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 2812	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		isset 2807	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylib 122	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2800

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-v 2802

This theorem is referenced by:  elex22  2816  elex2  2817  ceqsalt  2827  ceqsalg  2829  cgsexg  2836  cgsex2g  2837  cgsex4g  2838  vtoclgft  2852  vtocleg  2875  vtoclegft  2876  spc2egv  2894  spc2gv  2895  spc3egv  2896  spc3gv  2897  eqvincg  2928  tpid3g  3785  iinexgm  4242  copsex2t  4335  copsex2g  4336  ralxfr2d  4559  rexxfr2d  4560  fliftf  5935  eloprabga  6103  ovmpt4g  6139  spc2ed  6393  eroveu  6790  supelti  7192  genpassl  7734  genpassu  7735  eqord1  8653  nn1suc  9152  bj-inex  16438