Theorem elisset 2836

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 2833	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		isset 2828	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylib 122	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-v 2823

This theorem is used by:  elex22  2837  elex2  2838  ceqsalt  2848  ceqsalg  2850  cgsexg  2857  cgsex2g  2858  cgsex4g  2859  vtoclgft  2873  vtocleg  2896  vtoclegft  2897  spc2egv  2915  spc2gv  2916  spc3egv  2917  spc3gv  2918  eqvincg  2950  tpid3g  3828  iinexgm  4290  copsex2t  4385  copsex2g  4386  ralxfr2d  4610  rexxfr2d  4611  fliftf  6005  eloprabga  6175  ovmpt4g  6211  spc2ed  6469  eroveu  6900  supelti  7342  genpassl  7891  genpassu  7892  eqord1  8811  nn1suc  9323  bj-inex  16933