Theorem ssex 3997

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 3978 (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 3026	. 2 |- ( A C_ B <-> ( A i^i B ) = A )
2		ssex.1	. . . 4 |- B e. _V
3	2	inex2 3995	. . 3 |- ( A i^i B ) e. _V
4		eleq1 2157	. . 3 |- ( ( A i^i B ) = A -> ( ( A i^i B ) e. _V <-> A e. _V ) )
5	3, 4	mpbii 147	. 2 |- ( ( A i^i B ) = A -> A e. _V )
6	1, 5	sylbi 120	1 |- ( A C_ B -> A e. _V )

Syntax hints:   -> wi 4   = wceq 1296   e. wcel 1445   _Vcvv 2633   i^i cin 3012   C_ wss 3013

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019  df-ss 3026

This theorem is referenced by:  ssexi  3998  ssexg  3999  inteximm  4006  funimaexglem  5131  tfrexlem  6137  elinp  7130  negfi  10774  elcncf  12328