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Theorem disji 4011
Description: Property of a disjoint collection: if  B ( X )  =  C and  B ( Y )  =  D have a common element  Z, then  X  =  Y. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
disji.1  |-  ( x  =  X  ->  B  =  C )
disji.2  |-  ( x  =  Y  ->  B  =  D )
Assertion
Ref Expression
disji  |-  ( (Disj  x  e.  A B  /\  ( X  e.  A  /\  Y  e.  A
)  /\  ( Z  e.  C  /\  Z  e.  D ) )  ->  X  =  Y )
Distinct variable groups:    x, A    x, C    x, D    x, X    x, Y
Allowed substitution hints:    B( x)    Z( x)

Proof of Theorem disji
StepHypRef Expression
1 inelcm 3509 . 2  |-  ( ( Z  e.  C  /\  Z  e.  D )  ->  ( C  i^i  D
)  =/=  (/) )
2 disji.1 . . . . . 6  |-  ( x  =  X  ->  B  =  C )
3 disji.2 . . . . . 6  |-  ( x  =  Y  ->  B  =  D )
42, 3disji2 4010 . . . . 5  |-  ( (Disj  x  e.  A B  /\  ( X  e.  A  /\  Y  e.  A
)  /\  X  =/=  Y )  ->  ( C  i^i  D )  =  (/) )
543expia 1153 . . . 4  |-  ( (Disj  x  e.  A B  /\  ( X  e.  A  /\  Y  e.  A
) )  ->  ( X  =/=  Y  ->  ( C  i^i  D )  =  (/) ) )
65necon1d 2515 . . 3  |-  ( (Disj  x  e.  A B  /\  ( X  e.  A  /\  Y  e.  A
) )  ->  (
( C  i^i  D
)  =/=  (/)  ->  X  =  Y ) )
763impia 1148 . 2  |-  ( (Disj  x  e.  A B  /\  ( X  e.  A  /\  Y  e.  A
)  /\  ( C  i^i  D )  =/=  (/) )  ->  X  =  Y )
81, 7syl3an3 1217 1  |-  ( (Disj  x  e.  A B  /\  ( X  e.  A  /\  Y  e.  A
)  /\  ( Z  e.  C  /\  Z  e.  D ) )  ->  X  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    i^i cin 3151   (/)c0 3455  Disj wdisj 3993
This theorem is referenced by:  volfiniun  18904
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-reu 2550  df-rmo 2551  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-in 3159  df-nul 3456  df-disj 3994
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