Theorem ssex 4167

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 4148 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)

Hypothesis

Ref	Expression
ssex.1	|- B e. _V

Assertion

Ref	Expression
ssex	|- ( A C_ B -> A e. _V )

Proof of Theorem ssex

Step	Hyp	Ref	Expression
1		df-ss 3167	. 2 |- ( A C_ B <-> ( A i^i B ) = A )
2		ssex.1	. . . 4 |- B e. _V
3	2	inex2 4165	. . 3 |- ( A i^i B ) e. _V
4		eleq1 2256	. . 3 |- ( ( A i^i B ) = A -> ( ( A i^i B ) e. _V <-> A e. _V ) )
5	3, 4	mpbii 148	. 2 |- ( ( A i^i B ) = A -> A e. _V )
6	1, 5	sylbi 121	1 |- ( A C_ B -> A e. _V )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   _Vcvv 2760   i^i cin 3153   C_ wss 3154

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-sep 4148

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3160  df-ss 3167

This theorem is referenced by:  ssexi  4168  ssexg  4169  inteximm  4179  exmid1stab  4238  funimaexglem  5338  tfrexlem  6389  elinp  7536  suplocexprlem2b  7776  negfi  11374  ssomct  12605  ssnnctlemct  12606  nninfdc  12613  elcncf  14752  plyval  14911  sbthom  15586