Theorem ssexi 3954

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 3953	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
4	1, 3	ax-mp 7	1 ⊢ 𝐴 ∈ V

Syntax hints:   ∈ wcel 1436  Vcvv 2615   ⊆ wss 2988

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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-in 2994  df-ss 3001

This theorem is referenced by:  zfausab  3958  pp0ex  4000  ord3ex  4001  epse  4145  opabex  5484  oprabex  5858  phplem2  6523  phpm  6535  snexxph  6611  sbthlem2  6614  niex  6818  enqex  6866  enq0ex  6945  npex  6979  ltnqex  7055  gtnqex  7056  recexprlemell  7128  recexprlemelu  7129  enrex  7230  axcnex  7343  peano5nnnn  7374  reex  7423  nnex  8366  zex  8695  qex  9052  ixxex  9252  serclim0  10609  iserclim0  10610  climle  10638  prmex  11020