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Theorem disjiunOLD 4014
Description: A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (New usage 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 3995 . 2  |-  (Disj  x  e.  A B  <->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
2 disjiun 4013 . 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 459 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 358    /\ w3a 934   A.wal 1527    = wceq 1623    e. wcel 1684   E*wmo 2144    i^i cin 3151    C_ wss 3152   (/)c0 3455   U_ciun 3905  Disj wdisj 3993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rmo 2551  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-iun 3907  df-disj 3994
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