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Theorem disjiun2 37398
Description: In a disjoint collection, an indexed union is disjoint from an additional term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
disjiun2.1  |-  ( ph  -> Disj  x  e.  A  B
)
disjiun2.2  |-  ( ph  ->  C  C_  A )
disjiun2.3  |-  ( ph  ->  D  e.  ( A 
\  C ) )
disjiun2.4  |-  ( x  =  D  ->  B  =  E )
Assertion
Ref Expression
disjiun2  |-  ( ph  ->  ( U_ x  e.  C  B  i^i  E
)  =  (/) )
Distinct variable groups:    x, A    x, C    x, D    x, E
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem disjiun2
StepHypRef Expression
1 disjiun2.3 . . . 4  |-  ( ph  ->  D  e.  ( A 
\  C ) )
2 disjiun2.4 . . . . 5  |-  ( x  =  D  ->  B  =  E )
32iunxsng 4360 . . . 4  |-  ( D  e.  ( A  \  C )  ->  U_ x  e.  { D } B  =  E )
41, 3syl 17 . . 3  |-  ( ph  ->  U_ x  e.  { D } B  =  E )
54ineq2d 3634 . 2  |-  ( ph  ->  ( U_ x  e.  C  B  i^i  U_ x  e.  { D } B )  =  (
U_ x  e.  C  B  i^i  E ) )
6 disjiun2.1 . . 3  |-  ( ph  -> Disj  x  e.  A  B
)
7 disjiun2.2 . . 3  |-  ( ph  ->  C  C_  A )
8 eldifi 3555 . . . 4  |-  ( D  e.  ( A  \  C )  ->  D  e.  A )
9 snssi 4116 . . . 4  |-  ( D  e.  A  ->  { D }  C_  A )
101, 8, 93syl 18 . . 3  |-  ( ph  ->  { D }  C_  A )
111eldifbd 3417 . . . 4  |-  ( ph  ->  -.  D  e.  C
)
12 disjsn 4032 . . . 4  |-  ( ( C  i^i  { D } )  =  (/)  <->  -.  D  e.  C )
1311, 12sylibr 216 . . 3  |-  ( ph  ->  ( C  i^i  { D } )  =  (/) )
14 disjiun 4393 . . 3  |-  ( (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 )  =  (/) )
156, 7, 10, 13, 14syl13anc 1270 . 2  |-  ( ph  ->  ( U_ x  e.  C  B  i^i  U_ x  e.  { D } B )  =  (/) )
165, 15eqtr3d 2487 1  |-  ( ph  ->  ( U_ x  e.  C  B  i^i  E
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1444    e. wcel 1887    \ cdif 3401    i^i cin 3403    C_ wss 3404   (/)c0 3731   {csn 3968   U_ciun 4278  Disj wdisj 4373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rmo 2745  df-v 3047  df-sbc 3268  df-dif 3407  df-in 3411  df-ss 3418  df-nul 3732  df-sn 3969  df-iun 4280  df-disj 4374
This theorem is referenced by:  caratheodorylem1  38347
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