Theorem elisset 2633

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 2630	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		isset 2625	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylib 120	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   = wceq 1289  ∃wex 1426   ∈ wcel 1438  Vcvv 2619

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621

This theorem is referenced by:  elex22  2634  elex2  2635  ceqsalt  2645  ceqsalg  2647  cgsexg  2654  cgsex2g  2655  cgsex4g  2656  vtoclgft  2669  vtocleg  2690  vtoclegft  2691  spc2egv  2708  spc2gv  2709  spc3egv  2710  spc3gv  2711  eqvincg  2741  tpid3g  3555  iinexgm  3990  copsex2t  4072  copsex2g  4073  ralxfr2d  4286  rexxfr2d  4287  fliftf  5578  eloprabga  5735  ovmpt4g  5767  spc2ed  5998  eroveu  6381  supelti  6695  genpassl  7081  genpassu  7082  eqord1  7959  nn1suc  8439  bj-inex  11753