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Theorem riinn0 4157
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
StepHypRef Expression
1 incom 3525 . 2  |-  ( A  i^i  |^|_ x  e.  X  S )  =  (
|^|_ x  e.  X  S  i^i  A )
2 r19.2z 3709 . . . . 5  |-  ( ( X  =/=  (/)  /\  A. x  e.  X  S  C_  A )  ->  E. x  e.  X  S  C_  A
)
32ancoms 440 . . . 4  |-  ( ( A. x  e.  X  S  C_  A  /\  X  =/=  (/) )  ->  E. x  e.  X  S  C_  A
)
4 iinss 4134 . . . 4  |-  ( E. x  e.  X  S  C_  A  ->  |^|_ x  e.  X  S  C_  A
)
53, 4syl 16 . . 3  |-  ( ( A. x  e.  X  S  C_  A  /\  X  =/=  (/) )  ->  |^|_ x  e.  X  S  C_  A
)
6 df-ss 3326 . . 3  |-  ( |^|_ x  e.  X  S  C_  A 
<->  ( |^|_ x  e.  X  S  i^i  A )  = 
|^|_ x  e.  X  S )
75, 6sylib 189 . 2  |-  ( ( A. x  e.  X  S  C_  A  /\  X  =/=  (/) )  ->  ( |^|_ x  e.  X  S  i^i  A )  =  |^|_ x  e.  X  S )
81, 7syl5eq 2479 1  |-  ( ( A. x  e.  X  S  C_  A  /\  X  =/=  (/) )  ->  ( A  i^i  |^|_ x  e.  X  S )  =  |^|_ x  e.  X  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    =/= wne 2598   A.wral 2697   E.wrex 2698    i^i cin 3311    C_ wss 3312   (/)c0 3620   |^|_ciin 4086
This theorem is referenced by:  riinrab  4158  riiner  6969  mreriincl  13813  riinopn  16971  alexsublem  18065  fnemeet1  26349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621  df-iin 4088
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