Theorem elisset 2814

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 2811	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		isset 2806	. 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 2799

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 2801

This theorem is referenced by:  elex22  2815  elex2  2816  ceqsalt  2826  ceqsalg  2828  cgsexg  2835  cgsex2g  2836  cgsex4g  2837  vtoclgft  2851  vtocleg  2874  vtoclegft  2875  spc2egv  2893  spc2gv  2894  spc3egv  2895  spc3gv  2896  eqvincg  2927  tpid3g  3782  iinexgm  4238  copsex2t  4331  copsex2g  4332  ralxfr2d  4555  rexxfr2d  4556  fliftf  5929  eloprabga  6097  ovmpt4g  6133  spc2ed  6385  eroveu  6781  supelti  7180  genpassl  7722  genpassu  7723  eqord1  8641  nn1suc  9140  bj-inex  16325