Theorem elisset 2744

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 2741	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		isset 2736	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylib 121	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   = wceq 1348  ∃wex 1485   ∈ wcel 2141  Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by:  elex22  2745  elex2  2746  ceqsalt  2756  ceqsalg  2758  cgsexg  2765  cgsex2g  2766  cgsex4g  2767  vtoclgft  2780  vtocleg  2801  vtoclegft  2802  spc2egv  2820  spc2gv  2821  spc3egv  2822  spc3gv  2823  eqvincg  2854  tpid3g  3698  iinexgm  4140  copsex2t  4230  copsex2g  4231  ralxfr2d  4449  rexxfr2d  4450  fliftf  5778  eloprabga  5940  ovmpt4g  5975  spc2ed  6212  eroveu  6604  supelti  6979  genpassl  7486  genpassu  7487  eqord1  8402  nn1suc  8897  bj-inex  13942