Theorem elisset 2818

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 2815	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		isset 2810	. 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 2202  Vcvv 2803

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 2213

This theorem depends on definitions:  df-bi 117  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-v 2805

This theorem is referenced by:  elex22  2819  elex2  2820  ceqsalt  2830  ceqsalg  2832  cgsexg  2839  cgsex2g  2840  cgsex4g  2841  vtoclgft  2855  vtocleg  2878  vtoclegft  2879  spc2egv  2897  spc2gv  2898  spc3egv  2899  spc3gv  2900  eqvincg  2931  tpid3g  3791  iinexgm  4249  copsex2t  4343  copsex2g  4344  ralxfr2d  4567  rexxfr2d  4568  fliftf  5950  eloprabga  6118  ovmpt4g  6154  spc2ed  6407  eroveu  6838  supelti  7244  genpassl  7787  genpassu  7788  eqord1  8705  nn1suc  9204  bj-inex  16606