Theorem riinn0 4040

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 3448	. 2 |- (A i^i |^|_x e. X S) = (|^|_x e. X S i^i A)
2		r19.2z 3639	. . . . 5 |- ((X =/= (/) /\ A.x e. X S C_ A) -> E.x e. X S C_ A)
3	2	ancoms 439	. . . 4 |- ((A.x e. X S C_ A /\ X =/= (/)) -> E.x e. X S C_ A)
4		iinss 4017	. . . 4 |- (E.x e. X S C_ A -> |^|_x e. X S C_ A)
5	3, 4	syl 15	. . 3 |- ((A.x e. X S C_ A /\ X =/= (/)) -> |^|_x e. X S C_ A)
6		df-ss 3259	. . 3 |- (|^|_x e. X S C_ A <-> (|^|_x e. X S i^i A) = |^|_x e. X S)
7	5, 6	sylib 188	. 2 |- ((A.x e. X S C_ A /\ X =/= (/)) -> (|^|_x e. X S i^i A) = |^|_x e. X S)
8	1, 7	syl5eq 2397	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 358   = wceq 1642   =/= wne 2516  A.wral 2614  E.wrex 2615   i^i cin 3208   C_ wss 3257  (/)c0 3550  |^|_ciin 3970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551  df-iin 3972

This theorem is referenced by:  riinrab  4041