Theorem elisset 2613

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 2610	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		isset 2605	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylib 120	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   = wceq 1284  ∃wex 1421   ∈ wcel 1433  Vcvv 2601

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 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by:  elex22  2614  elex2  2615  ceqsalt  2625  ceqsalg  2627  cgsexg  2634  cgsex2g  2635  cgsex4g  2636  vtoclgft  2649  vtocleg  2669  vtoclegft  2670  spc2egv  2687  spc2gv  2688  spc3egv  2689  spc3gv  2690  eqvincg  2719  tpid3g  3505  iinexgm  3929  copsex2t  4000  copsex2g  4001  ralxfr2d  4214  rexxfr2d  4215  fliftf  5459  eloprabga  5611  ovmpt4g  5643  spc2ed  5874  eroveu  6220  supelti  6415  genpassl  6714  genpassu  6715  nn1suc  8058  bj-inex  10698