Theorem ssex 4224

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 4205 (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 3211	. 2 |- ( A C_ B <-> ( A i^i B ) = A )
2		ssex.1	. . . 4 |- B e. _V
3	2	inex2 4222	. . 3 |- ( A i^i B ) e. _V
4		eleq1 2292	. . 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 1395   e. wcel 2200   _Vcvv 2800   i^i cin 3197   C_ wss 3198

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-sep 4205

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-in 3204  df-ss 3211

This theorem is referenced by:  ssexi  4225  ssexg  4226  inteximm  4237  exmid1stab  4296  funimaexglem  5410  tfrexlem  6495  elinp  7684  suplocexprlem2b  7924  negfi  11779  ssomct  13056  ssnnctlemct  13057  nninfdc  13064  prdsval  13346  elcncf  15287  plyval  15446  sbthom  16566