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Theorem disjiunel 28254
Description: A set of elements B of a disjoint set A is disjoint with another element of that set. (Contributed by Thierry Arnoux, 24-May-2020.)
Hypotheses
Ref Expression
disjiunel.1  |-  ( ph  -> Disj  x  e.  A  B
)
disjiunel.2  |-  ( x  =  Y  ->  B  =  D )
disjiunel.3  |-  ( ph  ->  E  C_  A )
disjiunel.4  |-  ( ph  ->  Y  e.  ( A 
\  E ) )
Assertion
Ref Expression
disjiunel  |-  ( ph  ->  ( U_ x  e.  E  B  i^i  D
)  =  (/) )
Distinct variable groups:    x, A    x, D    x, E    x, Y
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem disjiunel
StepHypRef Expression
1 disjiunel.3 . . . . 5  |-  ( ph  ->  E  C_  A )
2 disjiunel.4 . . . . . . 7  |-  ( ph  ->  Y  e.  ( A 
\  E ) )
32eldifad 3427 . . . . . 6  |-  ( ph  ->  Y  e.  A )
43snssd 4129 . . . . 5  |-  ( ph  ->  { Y }  C_  A )
51, 4unssd 3621 . . . 4  |-  ( ph  ->  ( E  u.  { Y } )  C_  A
)
6 disjiunel.1 . . . 4  |-  ( ph  -> Disj  x  e.  A  B
)
7 disjss1 4392 . . . . 5  |-  ( ( E  u.  { Y } )  C_  A  ->  (Disj  x  e.  A  B  -> Disj  x  e.  ( E  u.  { Y }
) B ) )
87imp 435 . . . 4  |-  ( ( ( E  u.  { Y } )  C_  A  /\ Disj  x  e.  A  B
)  -> Disj  x  e.  ( E  u.  { Y } ) B )
95, 6, 8syl2anc 671 . . 3  |-  ( ph  -> Disj  x  e.  ( E  u.  { Y } ) B )
102eldifbd 3428 . . . 4  |-  ( ph  ->  -.  Y  e.  E
)
11 disjiunel.2 . . . . 5  |-  ( x  =  Y  ->  B  =  D )
1211disjunsn 28252 . . . 4  |-  ( ( Y  e.  A  /\  -.  Y  e.  E
)  ->  (Disj  x  e.  ( E  u.  { Y } ) B  <->  (Disj  x  e.  E  B  /\  ( U_ x  e.  E  B  i^i  D )  =  (/) ) ) )
133, 10, 12syl2anc 671 . . 3  |-  ( ph  ->  (Disj  x  e.  ( E  u.  { Y } ) B  <->  (Disj  x  e.  E  B  /\  ( U_ x  e.  E  B  i^i  D )  =  (/) ) ) )
149, 13mpbid 215 . 2  |-  ( ph  ->  (Disj  x  e.  E  B  /\  ( U_ x  e.  E  B  i^i  D )  =  (/) ) )
1514simprd 469 1  |-  ( ph  ->  ( U_ x  e.  E  B  i^i  D
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1454    e. wcel 1897    \ cdif 3412    u. cun 3413    i^i cin 3414    C_ wss 3415   (/)c0 3742   {csn 3979   U_ciun 4291  Disj wdisj 4386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-sn 3980  df-iun 4293  df-disj 4387
This theorem is referenced by:  disjuniel  28255
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