Theorem ssexi 3916

Description: The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)

Hypotheses

Ref	Expression
ssexi.1	⊢ 𝐵 ∈ V
ssexi.2	⊢ 𝐴 ⊆ 𝐵

Assertion

Ref	Expression
ssexi	⊢ 𝐴 ∈ V

Proof of Theorem ssexi

Step	Hyp	Ref	Expression
1		ssexi.2	. 2 ⊢ 𝐴 ⊆ 𝐵
2		ssexi.1	. . 3 ⊢ 𝐵 ∈ V
3	2	ssex 3915	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
4	1, 3	ax-mp 7	1 ⊢ 𝐴 ∈ V

Syntax hints:   ∈ wcel 1433  Vcvv 2601   ⊆ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986

This theorem is referenced by:  zfausab  3920  pp0ex  3960  ord3ex  3961  epse  4097  opabex  5406  oprabex  5775  phplem2  6339  phpm  6351  niex  6502  enqex  6550  enq0ex  6629  npex  6663  ltnqex  6739  gtnqex  6740  recexprlemell  6812  recexprlemelu  6813  enrex  6914  axcnex  7027  peano5nnnn  7058  reex  7107  nnex  8045  zex  8360  qex  8717  ixxex  8922  frecuzrdgrrn  9410  frec2uzrdg  9411  frecuzrdgrom  9412  frecuzrdgsuc  9417  resqrexlemf  9893  resqrexlemf1  9894  resqrexlemfp1  9895  iserclim0  10144  climle  10172  prmex  10495