MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  riinn0 Structured version   Unicode version

Theorem riinn0 4395
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 3686 . 2  |-  ( A  i^i  |^|_ x  e.  X  S )  =  (
|^|_ x  e.  X  S  i^i  A )
2 r19.2z 3912 . . . . 5  |-  ( ( X  =/=  (/)  /\  A. x  e.  X  S  C_  A )  ->  E. x  e.  X  S  C_  A
)
32ancoms 453 . . . 4  |-  ( ( A. x  e.  X  S  C_  A  /\  X  =/=  (/) )  ->  E. x  e.  X  S  C_  A
)
4 iinss 4371 . . . 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 3485 . . 3  |-  ( |^|_ x  e.  X  S  C_  A 
<->  ( |^|_ x  e.  X  S  i^i  A )  = 
|^|_ x  e.  X  S )
75, 6sylib 196 . 2  |-  ( ( A. x  e.  X  S  C_  A  /\  X  =/=  (/) )  ->  ( |^|_ x  e.  X  S  i^i  A )  =  |^|_ x  e.  X  S )
81, 7syl5eq 2515 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 369    = wceq 1374    =/= wne 2657   A.wral 2809   E.wrex 2810    i^i cin 3470    C_ wss 3471   (/)c0 3780   |^|_ciin 4321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-v 3110  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3781  df-iin 4323
This theorem is referenced by:  riinrab  4396  riiner  7376  mreriincl  14844  riinopn  19179  alexsublem  20274  fnemeet1  29776
  Copyright terms: Public domain W3C validator