Theorem elisset 2562

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 2560	. 2 ⊢ (A ∈ 𝑉 → A ∈ V)
2		isset 2555	. 2 ⊢ (A ∈ V ↔ ∃x x = A)
3	1, 2	sylib 127	1 ⊢ (A ∈ 𝑉 → ∃x x = A)

Syntax hints:   → wi 4   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553

This theorem is referenced by:  elex22  2563  elex2  2564  ceqsalt  2574  ceqsalg  2576  cgsexg  2583  cgsex2g  2584  cgsex4g  2585  vtoclgft  2598  vtocleg  2618  vtoclegft  2619  spc2egv  2636  spc2gv  2637  spc3egv  2638  spc3gv  2639  eqvincg  2662  tpid3g  3474  iinexgm  3899  copsex2t  3973  copsex2g  3974  ralxfr2d  4162  rexxfr2d  4163  fliftf  5382  eloprabga  5533  ovmpt4g  5565  eroveu  6133  genpassl  6507  genpassu  6508  nn1suc  7714  bj-inex  9362