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Theorem disjprg 4149
Description: A pair collection is disjoint iff the two sets in the family have empty intersection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
disjprg.1  |-  ( x  =  A  ->  C  =  D )
disjprg.2  |-  ( x  =  B  ->  C  =  E )
Assertion
Ref Expression
disjprg  |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B )  -> 
(Disj  x  e.  { A ,  B } C  <->  ( D  i^i  E )  =  (/) ) )
Distinct variable groups:    x, A    x, B    x, D    x, E
Allowed substitution hints:    C( x)    V( x)

Proof of Theorem disjprg
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2393 . . . . . . 7  |-  ( y  =  A  ->  (
y  =  z  <->  A  =  z ) )
2 nfcv 2523 . . . . . . . . . 10  |-  F/_ x A
3 nfcv 2523 . . . . . . . . . 10  |-  F/_ x D
4 disjprg.1 . . . . . . . . . 10  |-  ( x  =  A  ->  C  =  D )
52, 3, 4csbhypf 3229 . . . . . . . . 9  |-  ( y  =  A  ->  [_ y  /  x ]_ C  =  D )
65ineq1d 3484 . . . . . . . 8  |-  ( y  =  A  ->  ( [_ y  /  x ]_ C  i^i  [_ z  /  x ]_ C )  =  ( D  i^i  [_ z  /  x ]_ C ) )
76eqeq1d 2395 . . . . . . 7  |-  ( y  =  A  ->  (
( [_ y  /  x ]_ C  i^i  [_ z  /  x ]_ C )  =  (/)  <->  ( D  i^i  [_ z  /  x ]_ C )  =  (/) ) )
81, 7orbi12d 691 . . . . . 6  |-  ( y  =  A  ->  (
( y  =  z  \/  ( [_ y  /  x ]_ C  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<->  ( A  =  z  \/  ( D  i^i  [_ z  /  x ]_ C )  =  (/) ) ) )
98ralbidv 2669 . . . . 5  |-  ( y  =  A  ->  ( A. z  e.  { A ,  B }  ( y  =  z  \/  ( [_ y  /  x ]_ C  i^i  [_ z  /  x ]_ C )  =  (/) )  <->  A. z  e.  { A ,  B }  ( A  =  z  \/  ( D  i^i  [_ z  /  x ]_ C )  =  (/) ) ) )
10 eqeq1 2393 . . . . . . 7  |-  ( y  =  B  ->  (
y  =  z  <->  B  =  z ) )
11 nfcv 2523 . . . . . . . . . 10  |-  F/_ x B
12 nfcv 2523 . . . . . . . . . 10  |-  F/_ x E
13 disjprg.2 . . . . . . . . . 10  |-  ( x  =  B  ->  C  =  E )
1411, 12, 13csbhypf 3229 . . . . . . . . 9  |-  ( y  =  B  ->  [_ y  /  x ]_ C  =  E )
1514ineq1d 3484 . . . . . . . 8  |-  ( y  =  B  ->  ( [_ y  /  x ]_ C  i^i  [_ z  /  x ]_ C )  =  ( E  i^i  [_ z  /  x ]_ C ) )
1615eqeq1d 2395 . . . . . . 7  |-  ( y  =  B  ->  (
( [_ y  /  x ]_ C  i^i  [_ z  /  x ]_ C )  =  (/)  <->  ( E  i^i  [_ z  /  x ]_ C )  =  (/) ) )
1710, 16orbi12d 691 . . . . . 6  |-  ( y  =  B  ->  (
( y  =  z  \/  ( [_ y  /  x ]_ C  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<->  ( B  =  z  \/  ( E  i^i  [_ z  /  x ]_ C )  =  (/) ) ) )
1817ralbidv 2669 . . . . 5  |-  ( y  =  B  ->  ( A. z  e.  { A ,  B }  ( y  =  z  \/  ( [_ y  /  x ]_ C  i^i  [_ z  /  x ]_ C )  =  (/) )  <->  A. z  e.  { A ,  B }  ( B  =  z  \/  ( E  i^i  [_ z  /  x ]_ C )  =  (/) ) ) )
199, 18ralprg 3800 . . . 4  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( A. y  e. 
{ A ,  B } A. z  e.  { A ,  B } 
( y  =  z  \/  ( [_ y  /  x ]_ C  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<->  ( A. z  e. 
{ A ,  B }  ( A  =  z  \/  ( D  i^i  [_ z  /  x ]_ C )  =  (/) )  /\  A. z  e. 
{ A ,  B }  ( B  =  z  \/  ( E  i^i  [_ z  /  x ]_ C )  =  (/) ) ) ) )
20193adant3 977 . . 3  |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B )  -> 
( A. y  e. 
{ A ,  B } A. z  e.  { A ,  B } 
( y  =  z  \/  ( [_ y  /  x ]_ C  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<->  ( A. z  e. 
{ A ,  B }  ( A  =  z  \/  ( D  i^i  [_ z  /  x ]_ C )  =  (/) )  /\  A. z  e. 
{ A ,  B }  ( B  =  z  \/  ( E  i^i  [_ z  /  x ]_ C )  =  (/) ) ) ) )
21 id 20 . . . . . . . . . 10  |-  ( z  =  A  ->  z  =  A )
2221eqcomd 2392 . . . . . . . . 9  |-  ( z  =  A  ->  A  =  z )
2322orcd 382 . . . . . . . 8  |-  ( z  =  A  ->  ( A  =  z  \/  ( D  i^i  [_ z  /  x ]_ C )  =  (/) ) )
24 a1tru 1336 . . . . . . . 8  |-  ( z  =  A  ->  T.  )
2523, 242thd 232 . . . . . . 7  |-  ( z  =  A  ->  (
( A  =  z  \/  ( D  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<->  T.  ) )
26 eqeq2 2396 . . . . . . . 8  |-  ( z  =  B  ->  ( A  =  z  <->  A  =  B ) )
2711, 12, 13csbhypf 3229 . . . . . . . . . 10  |-  ( z  =  B  ->  [_ z  /  x ]_ C  =  E )
2827ineq2d 3485 . . . . . . . . 9  |-  ( z  =  B  ->  ( D  i^i  [_ z  /  x ]_ C )  =  ( D  i^i  E ) )
2928eqeq1d 2395 . . . . . . . 8  |-  ( z  =  B  ->  (
( D  i^i  [_ z  /  x ]_ C )  =  (/)  <->  ( D  i^i  E )  =  (/) ) )
3026, 29orbi12d 691 . . . . . . 7  |-  ( z  =  B  ->  (
( A  =  z  \/  ( D  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<->  ( A  =  B  \/  ( D  i^i  E )  =  (/) ) ) )
3125, 30ralprg 3800 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( A. z  e. 
{ A ,  B }  ( A  =  z  \/  ( D  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<->  (  T.  /\  ( A  =  B  \/  ( D  i^i  E )  =  (/) ) ) ) )
32313adant3 977 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B )  -> 
( A. z  e. 
{ A ,  B }  ( A  =  z  \/  ( D  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<->  (  T.  /\  ( A  =  B  \/  ( D  i^i  E )  =  (/) ) ) ) )
33 simp3 959 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B )  ->  A  =/=  B )
3433neneqd 2566 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B )  ->  -.  A  =  B
)
35 biorf 395 . . . . . . 7  |-  ( -.  A  =  B  -> 
( ( D  i^i  E )  =  (/)  <->  ( A  =  B  \/  ( D  i^i  E )  =  (/) ) ) )
3634, 35syl 16 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B )  -> 
( ( D  i^i  E )  =  (/)  <->  ( A  =  B  \/  ( D  i^i  E )  =  (/) ) ) )
37 tru 1327 . . . . . . 7  |-  T.
3837biantrur 493 . . . . . 6  |-  ( ( A  =  B  \/  ( D  i^i  E )  =  (/) )  <->  (  T.  /\  ( A  =  B  \/  ( D  i^i  E )  =  (/) ) ) )
3936, 38syl6bb 253 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B )  -> 
( ( D  i^i  E )  =  (/)  <->  (  T.  /\  ( A  =  B  \/  ( D  i^i  E )  =  (/) ) ) ) )
4032, 39bitr4d 248 . . . 4  |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B )  -> 
( A. z  e. 
{ A ,  B }  ( A  =  z  \/  ( D  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<->  ( D  i^i  E
)  =  (/) ) )
41 eqeq2 2396 . . . . . . . . 9  |-  ( z  =  A  ->  ( B  =  z  <->  B  =  A ) )
42 eqcom 2389 . . . . . . . . 9  |-  ( B  =  A  <->  A  =  B )
4341, 42syl6bb 253 . . . . . . . 8  |-  ( z  =  A  ->  ( B  =  z  <->  A  =  B ) )
442, 3, 4csbhypf 3229 . . . . . . . . . . 11  |-  ( z  =  A  ->  [_ z  /  x ]_ C  =  D )
4544ineq2d 3485 . . . . . . . . . 10  |-  ( z  =  A  ->  ( E  i^i  [_ z  /  x ]_ C )  =  ( E  i^i  D ) )
46 incom 3476 . . . . . . . . . 10  |-  ( E  i^i  D )  =  ( D  i^i  E
)
4745, 46syl6eq 2435 . . . . . . . . 9  |-  ( z  =  A  ->  ( E  i^i  [_ z  /  x ]_ C )  =  ( D  i^i  E ) )
4847eqeq1d 2395 . . . . . . . 8  |-  ( z  =  A  ->  (
( E  i^i  [_ z  /  x ]_ C )  =  (/)  <->  ( D  i^i  E )  =  (/) ) )
4943, 48orbi12d 691 . . . . . . 7  |-  ( z  =  A  ->  (
( B  =  z  \/  ( E  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<->  ( A  =  B  \/  ( D  i^i  E )  =  (/) ) ) )
50 id 20 . . . . . . . . . 10  |-  ( z  =  B  ->  z  =  B )
5150eqcomd 2392 . . . . . . . . 9  |-  ( z  =  B  ->  B  =  z )
5251orcd 382 . . . . . . . 8  |-  ( z  =  B  ->  ( B  =  z  \/  ( E  i^i  [_ z  /  x ]_ C )  =  (/) ) )
53 a1tru 1336 . . . . . . . 8  |-  ( z  =  B  ->  T.  )
5452, 532thd 232 . . . . . . 7  |-  ( z  =  B  ->  (
( B  =  z  \/  ( E  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<->  T.  ) )
5549, 54ralprg 3800 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( A. z  e. 
{ A ,  B }  ( B  =  z  \/  ( E  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<->  ( ( A  =  B  \/  ( D  i^i  E )  =  (/) )  /\  T.  )
) )
56553adant3 977 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B )  -> 
( A. z  e. 
{ A ,  B }  ( B  =  z  \/  ( E  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<->  ( ( A  =  B  \/  ( D  i^i  E )  =  (/) )  /\  T.  )
) )
5737biantru 492 . . . . . 6  |-  ( ( A  =  B  \/  ( D  i^i  E )  =  (/) )  <->  ( ( A  =  B  \/  ( D  i^i  E )  =  (/) )  /\  T.  ) )
5836, 57syl6bb 253 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B )  -> 
( ( D  i^i  E )  =  (/)  <->  ( ( A  =  B  \/  ( D  i^i  E )  =  (/) )  /\  T.  ) ) )
5956, 58bitr4d 248 . . . 4  |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B )  -> 
( A. z  e. 
{ A ,  B }  ( B  =  z  \/  ( E  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<->  ( D  i^i  E
)  =  (/) ) )
6040, 59anbi12d 692 . . 3  |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B )  -> 
( ( A. z  e.  { A ,  B }  ( A  =  z  \/  ( D  i^i  [_ z  /  x ]_ C )  =  (/) )  /\  A. z  e. 
{ A ,  B }  ( B  =  z  \/  ( E  i^i  [_ z  /  x ]_ C )  =  (/) ) )  <->  ( ( D  i^i  E )  =  (/)  /\  ( D  i^i  E )  =  (/) ) ) )
6120, 60bitrd 245 . 2  |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B )  -> 
( A. y  e. 
{ A ,  B } A. z  e.  { A ,  B } 
( y  =  z  \/  ( [_ y  /  x ]_ C  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<->  ( ( D  i^i  E )  =  (/)  /\  ( D  i^i  E )  =  (/) ) ) )
62 disjors 4139 . 2  |-  (Disj  x  e.  { A ,  B } C  <->  A. y  e.  { A ,  B } A. z  e.  { A ,  B }  ( y  =  z  \/  ( [_ y  /  x ]_ C  i^i  [_ z  /  x ]_ C )  =  (/) ) )
63 pm4.24 625 . 2  |-  ( ( D  i^i  E )  =  (/)  <->  ( ( D  i^i  E )  =  (/)  /\  ( D  i^i  E )  =  (/) ) )
6461, 62, 633bitr4g 280 1  |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B )  -> 
(Disj  x  e.  { A ,  B } C  <->  ( D  i^i  E )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    T. wtru 1322    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   [_csb 3194    i^i cin 3262   (/)c0 3571   {cpr 3758  Disj wdisj 4123
This theorem is referenced by:  disjdifprg  23861  probun  24456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-nul 3572  df-sn 3763  df-pr 3764  df-disj 4124
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