Theorem ssex 4189

Description: The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22. This is one way to express the Axiom of Separation ax-sep 4170 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)

Hypothesis

Ref	Expression
ssex.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
ssex	⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)

Proof of Theorem ssex

Step	Hyp	Ref	Expression
1		df-ss 3183	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
2		ssex.1	. . . 4 ⊢ 𝐵 ∈ V
3	2	inex2 4187	. . 3 ⊢ (𝐴 ∩ 𝐵) ∈ V
4		eleq1 2269	. . 3 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 → ((𝐴 ∩ 𝐵) ∈ V ↔ 𝐴 ∈ V))
5	3, 4	mpbii 148	. 2 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 → 𝐴 ∈ V)
6	1, 5	sylbi 121	1 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2177  Vcvv 2773   ∩ cin 3169   ⊆ wss 3170

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-sep 4170

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176  df-ss 3183

This theorem is referenced by:  ssexi  4190  ssexg  4191  inteximm  4201  exmid1stab  4260  funimaexglem  5366  tfrexlem  6433  elinp  7607  suplocexprlem2b  7847  negfi  11614  ssomct  12891  ssnnctlemct  12892  nninfdc  12899  prdsval  13180  elcncf  15120  plyval  15279  sbthom  16106