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Theorem riinn0 4372
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 3656 . 2  |-  ( A  i^i  |^|_ x  e.  X  S )  =  (
|^|_ x  e.  X  S  i^i  A )
2 r19.2z 3887 . . . . 5  |-  ( ( X  =/=  (/)  /\  A. x  e.  X  S  C_  A )  ->  E. x  e.  X  S  C_  A
)
32ancoms 455 . . . 4  |-  ( ( A. x  e.  X  S  C_  A  /\  X  =/=  (/) )  ->  E. x  e.  X  S  C_  A
)
4 iinss 4348 . . . 4  |-  ( E. x  e.  X  S  C_  A  ->  |^|_ x  e.  X  S  C_  A
)
53, 4syl 17 . . 3  |-  ( ( A. x  e.  X  S  C_  A  /\  X  =/=  (/) )  ->  |^|_ x  e.  X  S  C_  A
)
6 df-ss 3451 . . 3  |-  ( |^|_ x  e.  X  S  C_  A 
<->  ( |^|_ x  e.  X  S  i^i  A )  = 
|^|_ x  e.  X  S )
75, 6sylib 200 . 2  |-  ( ( A. x  e.  X  S  C_  A  /\  X  =/=  (/) )  ->  ( |^|_ x  e.  X  S  i^i  A )  =  |^|_ x  e.  X  S )
81, 7syl5eq 2476 1  |-  ( ( A. x  e.  X  S  C_  A  /\  X  =/=  (/) )  ->  ( A  i^i  |^|_ x  e.  X  S )  =  |^|_ x  e.  X  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1438    =/= wne 2619   A.wral 2776   E.wrex 2777    i^i cin 3436    C_ wss 3437   (/)c0 3762   |^|_ciin 4298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-v 3084  df-dif 3440  df-in 3444  df-ss 3451  df-nul 3763  df-iin 4300
This theorem is referenced by:  riinrab  4373  riiner  7442  mreriincl  15497  riinopn  19930  alexsublem  21051  fnemeet1  31021
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