Theorem elisset 2786

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 2783	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		isset 2778	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylib 122	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

Syntax hints:   → wi 4   = wceq 1373  ∃wex 1515   ∈ wcel 2176  Vcvv 2772

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-v 2774

This theorem is referenced by:  elex22  2787  elex2  2788  ceqsalt  2798  ceqsalg  2800  cgsexg  2807  cgsex2g  2808  cgsex4g  2809  vtoclgft  2823  vtocleg  2844  vtoclegft  2845  spc2egv  2863  spc2gv  2864  spc3egv  2865  spc3gv  2866  eqvincg  2897  tpid3g  3748  iinexgm  4198  copsex2t  4289  copsex2g  4290  ralxfr2d  4511  rexxfr2d  4512  fliftf  5868  eloprabga  6032  ovmpt4g  6068  spc2ed  6319  eroveu  6713  supelti  7104  genpassl  7637  genpassu  7638  eqord1  8556  nn1suc  9055  bj-inex  15843