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Theorem disjif 26072
Description: Property of a disjoint collection: if  B ( x ) and  B ( Y )  =  D have a common element  Z, then  x  =  Y. (Contributed by Thierry Arnoux, 30-Dec-2016.)
Hypotheses
Ref Expression
disjif.1  |-  F/_ x C
disjif.2  |-  ( x  =  Y  ->  B  =  C )
Assertion
Ref Expression
disjif  |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A
)  /\  ( Z  e.  B  /\  Z  e.  C ) )  ->  x  =  Y )
Distinct variable groups:    x, A    x, Y
Allowed substitution hints:    B( x)    C( x)    Z( x)

Proof of Theorem disjif
StepHypRef Expression
1 inelcm 3840 . 2  |-  ( ( Z  e.  B  /\  Z  e.  C )  ->  ( B  i^i  C
)  =/=  (/) )
2 disjif.1 . . . . . 6  |-  F/_ x C
3 disjif.2 . . . . . 6  |-  ( x  =  Y  ->  B  =  C )
42, 3disji2f 26071 . . . . 5  |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A
)  /\  x  =/=  Y )  ->  ( B  i^i  C )  =  (/) )
543expia 1190 . . . 4  |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A
) )  ->  (
x  =/=  Y  -> 
( B  i^i  C
)  =  (/) ) )
65necon1d 2676 . . 3  |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A
) )  ->  (
( B  i^i  C
)  =/=  (/)  ->  x  =  Y ) )
763impia 1185 . 2  |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A
)  /\  ( B  i^i  C )  =/=  (/) )  ->  x  =  Y )
81, 7syl3an3 1254 1  |-  ( (Disj  x  e.  A  B  /\  ( x  e.  A  /\  Y  e.  A
)  /\  ( Z  e.  B  /\  Z  e.  C ) )  ->  x  =  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   F/_wnfc 2602    =/= wne 2647    i^i cin 3434   (/)c0 3744  Disj wdisj 4369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-in 3442  df-nul 3745  df-disj 4370
This theorem is referenced by:  disjabrex  26076
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