Theorem ssin 3429

Description: Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26. (Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
ssin	|- ( ( A C_ B /\ A C_ C ) <-> A C_ ( B i^i C ) )

Proof of Theorem ssin

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3390	. . . . 5 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
2	1	imbi2i 226	. . . 4 |- ( ( x e. A -> x e. ( B i^i C ) ) <-> ( x e. A -> ( x e. B /\ x e. C ) ) )
3	2	albii 1518	. . 3 |- ( A. x ( x e. A -> x e. ( B i^i C ) ) <-> A. x ( x e. A -> ( x e. B /\ x e. C ) ) )
4		jcab 607	. . . 4 |- ( ( x e. A -> ( x e. B /\ x e. C ) ) <-> ( ( x e. A -> x e. B ) /\ ( x e. A -> x e. C ) ) )
5	4	albii 1518	. . 3 |- ( A. x ( x e. A -> ( x e. B /\ x e. C ) ) <-> A. x ( ( x e. A -> x e. B ) /\ ( x e. A -> x e. C ) ) )
6		19.26 1529	. . 3 |- ( A. x ( ( x e. A -> x e. B ) /\ ( x e. A -> x e. C ) ) <-> ( A. x ( x e. A -> x e. B ) /\ A. x ( x e. A -> x e. C ) ) )
7	3, 5, 6	3bitrri 207	. 2 |- ( ( A. x ( x e. A -> x e. B ) /\ A. x ( x e. A -> x e. C ) ) <-> A. x ( x e. A -> x e. ( B i^i C ) ) )
8		ssalel 3215	. . 3 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
9		ssalel 3215	. . 3 |- ( A C_ C <-> A. x ( x e. A -> x e. C ) )
10	8, 9	anbi12i 460	. 2 |- ( ( A C_ B /\ A C_ C ) <-> ( A. x ( x e. A -> x e. B ) /\ A. x ( x e. A -> x e. C ) ) )
11		ssalel 3215	. 2 |- ( A C_ ( B i^i C ) <-> A. x ( x e. A -> x e. ( B i^i C ) ) )
12	7, 10, 11	3bitr4i 212	1 |- ( ( A C_ B /\ A C_ C ) <-> A C_ ( B i^i C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1395   e. wcel 2202   i^i cin 3199   C_ wss 3200

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213

This theorem is referenced by:  ssini  3430  ssind  3431  uneqin  3458  trin  4197  pwin  4379  peano5  4696  fin  5523  tgval  13344  eltg3i  14779  innei  14886  cnptoprest2  14963