Theorem iinss 4381

Description: Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iinss

Distinct variable group:   ,

Proof of Theorem iinss

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3112	. . . 4
2		eliin 4336	. . . 4
3	1, 2	ax-mp 5	. . 3
4		ssel 3497	. . . . 5
5	4	reximi 2925	. . . 4
6		r19.36v 3005	. . . 4
7	5, 6	syl 16	. . 3
8	3, 7	syl5bi 217	. 2
9	8	ssrdv 3509	1

Syntax hints:  ->wi 4  <->wb 184  e.wcel 1818  A.wral 2807  E.wrex 2808  cvv 3109  C_wss 3475  |^|_ciin 4331

This theorem is referenced by:  riinn0  4405  reliin  5129  cnviin  5549  iiner  7402  scott0  8325  cfslb  8667  ptbasfi  20082  iscmet3  21732  fnemeet1  30184  pmapglb2N  35495  pmapglb2xN  35496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-in 3482  df-ss 3489  df-iin 4333