Theorem elexd 2694

Description: If a class is a member of another class, it is a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypothesis

Ref	Expression
elexd.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
elexd	⊢ (𝜑 → 𝐴 ∈ V)

Proof of Theorem elexd

Step	Hyp	Ref	Expression
1		elexd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		elex 2692	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
3	1, 2	syl 14	1 ⊢ (𝜑 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1480  Vcvv 2681

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-v 2683

This theorem is referenced by:  dmmptd  5248  tfr1onlemsucfn  6230  tfrcllemsucfn  6243  frecrdg  6298  unsnfidcel  6802  fnfi  6818  caseinl  6969  caseinr  6970  acfun  7056  seq3val  10224  seqvalcd  10225  hashennn  10519  lcmval  11733  hashdvds  11886  ennnfonelemp1  11908  isstruct2r  11959  strnfvnd  11968  strfvssn  11970  strslfv2d  11990  setsslid  11998  ressid2  12007  ressval2  12008  istopon  12169  istps  12188  tgclb  12223  restbasg  12326  restco  12332  lmfval  12350  cnfval  12352  cnpfval  12353  cnpval  12356  txcnp  12429  txrest  12434  ismet2  12512  xmetpsmet  12527  mopnval  12600  comet  12657  reldvg  12806  dvmptclx  12838