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Theorem disjiunOLD 4144
Description: A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
disjiunOLD  |-  ( ( A. y E* x
( x  e.  A  /\  y  e.  B
)  /\  ( C  C_  A  /\  D  C_  A  /\  ( C  i^i  D )  =  (/) ) )  ->  ( U_ x  e.  C  B  i^i  U_ x  e.  D  B
)  =  (/) )
Distinct variable groups:    x, y, A    y, B    x, C, y    x, D, y
Allowed substitution hint:    B( x)

Proof of Theorem disjiunOLD
StepHypRef Expression
1 dfdisj2 4125 . 2  |-  (Disj  x  e.  A B  <->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
2 disjiun 4143 . 2  |-  ( (Disj  x  e.  A B  /\  ( C  C_  A  /\  D  C_  A  /\  ( C  i^i  D )  =  (/) ) )  -> 
( U_ x  e.  C  B  i^i  U_ x  e.  D  B )  =  (/) )
31, 2sylanbr 460 1  |-  ( ( A. y E* x
( x  e.  A  /\  y  e.  B
)  /\  ( C  C_  A  /\  D  C_  A  /\  ( C  i^i  D )  =  (/) ) )  ->  ( U_ x  e.  C  B  i^i  U_ x  e.  D  B
)  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   A.wal 1546    = wceq 1649    e. wcel 1717   E*wmo 2239    i^i cin 3262    C_ wss 3263   (/)c0 3571   U_ciun 4035  Disj wdisj 4123
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rmo 2657  df-v 2901  df-dif 3266  df-in 3270  df-ss 3277  df-nul 3572  df-iun 4037  df-disj 4124
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