Theorem ssex 4142

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 4123 (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 3144	. 2 |- ( A C_ B <-> ( A i^i B ) = A )
2		ssex.1	. . . 4 |- B e. _V
3	2	inex2 4140	. . 3 |- ( A i^i B ) e. _V
4		eleq1 2240	. . 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 1353   e. wcel 2148   _Vcvv 2739   i^i cin 3130   C_ wss 3131

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4123

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-ss 3144

This theorem is referenced by:  ssexi  4143  ssexg  4144  inteximm  4151  exmid1stab  4210  funimaexglem  5301  tfrexlem  6337  elinp  7475  suplocexprlem2b  7715  negfi  11238  ssomct  12448  ssnnctlemct  12449  nninfdc  12456  elcncf  14145  sbthom  14859