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  3781  iinexgm  4237  copsex2t  4330  copsex2g  4331  ralxfr2d  4554  rexxfr2d  4555  fliftf  5922  eloprabga  6090  ovmpt4g  6126  spc2ed  6377  eroveu  6771  supelti  7165  genpassl  7707  genpassu  7708  eqord1  8626  nn1suc  9125  bj-inex  16228