Theorem riinn0 4595

Description: Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
riinn0	|- ( ( A. x e. X S C_ A /\ X =/= (/) ) -> ( A i^i |^|_ x e. X S ) = |^|_ x e. X S )

Distinct variable groups:   x, A   x, X

Allowed substitution hint:   S( x)

Proof of Theorem riinn0

Step	Hyp	Ref	Expression
1		incom 3805	. 2 |- ( A i^i |^|_ x e. X S ) = ( |^|_ x e. X S i^i A )
2		r19.2z 4060	. . . . 5 |- ( ( X =/= (/) /\ A. x e. X S C_ A ) -> E. x e. X S C_ A )
3	2	ancoms 469	. . . 4 |- ( ( A. x e. X S C_ A /\ X =/= (/) ) -> E. x e. X S C_ A )
4		iinss 4571	. . . 4 |- ( E. x e. X S C_ A -> |^|_ x e. X S C_ A )
5	3, 4	syl 17	. . 3 |- ( ( A. x e. X S C_ A /\ X =/= (/) ) -> |^|_ x e. X S C_ A )
6		df-ss 3588	. . 3 |- ( |^|_ x e. X S C_ A <-> ( |^|_ x e. X S i^i A ) = |^|_ x e. X S )
7	5, 6	sylib 208	. 2 |- ( ( A. x e. X S C_ A /\ X =/= (/) ) -> ( |^|_ x e. X S i^i A ) = |^|_ x e. X S )
8	1, 7	syl5eq 2668	1 |- ( ( A. x e. X S C_ A /\ X =/= (/) ) -> ( A i^i |^|_ x e. X S ) = |^|_ x e. X S )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   =/= wne 2794   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   (/)c0 3915   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-iin 4523

This theorem is referenced by:  riinrab  4596  riiner  7820  mreriincl  16258  riinopn  20713  alexsublem  21848  fnemeet1  32361