Theorem elexi 2775

Description: If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.)

Hypothesis

Ref	Expression
elisseti.1	⊢ 𝐴 ∈ 𝐵

Assertion

Ref	Expression
elexi	⊢ 𝐴 ∈ V

Proof of Theorem elexi

Step	Hyp	Ref	Expression
1		elisseti.1	. 2 ⊢ 𝐴 ∈ 𝐵
2		elex 2774	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ V

Syntax hints:   ∈ wcel 2167  Vcvv 2763

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-v 2765

This theorem is referenced by:  elpwi2  4192  onunisuci  4468  ordsoexmid  4599  1oex  6491  fnoei  6519  oeiexg  6520  endisj  6892  unfiexmid  6988  snexxph  7025  djuex  7118  0ct  7182  nninfex  7196  infnninfOLD  7200  nnnninf  7201  ctssexmid  7225  nninfdcinf  7246  nninfwlporlem  7248  nninfwlpoimlemg  7250  pm54.43  7269  pw1ne3  7313  3nsssucpw1  7319  2omotaplemst  7341  prarloclemarch2  7503  opelreal  7911  elreal  7912  elreal2  7914  eqresr  7920  c0ex  8037  1ex  8038  pnfex  8097  sup3exmid  9001  2ex  9079  3ex  9083  elxr  9868  xnn0nnen  10546  setsslid  12754  setsslnid  12755  prdsex  12971  rmodislmod  13983  fnpsr  14297  lgsdir2lem3  15355  2omapen  15727  subctctexmid  15731  0nninf  15735  nninffeq  15751