Theorem ssex 3894

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 3875 (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 2931	. 2 |- ( A C_ B <-> ( A i^i B ) = A )
2		ssex.1	. . . 4 |- B e. _V
3	2	inex2 3892	. . 3 |- ( A i^i B ) e. _V
4		eleq1 2100	. . 3 |- ( ( A i^i B ) = A -> ( ( A i^i B ) e. _V <-> A e. _V ) )
5	3, 4	mpbii 136	. 2 |- ( ( A i^i B ) = A -> A e. _V )
6	1, 5	sylbi 114	1 |- ( A C_ B -> A e. _V )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   _Vcvv 2557   i^i cin 2916   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931

This theorem is referenced by:  ssexi  3895  ssexg  3896  inteximm  3903  funimaexglem  4982  tfrexlem  5948  elinp  6572