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Theorem disjprg 4435
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 2458 . . . . . . 7  |-  ( y  =  A  ->  (
y  =  z  <->  A  =  z ) )
2 nfcv 2616 . . . . . . . . . 10  |-  F/_ x A
3 nfcv 2616 . . . . . . . . . 10  |-  F/_ x D
4 disjprg.1 . . . . . . . . . 10  |-  ( x  =  A  ->  C  =  D )
52, 3, 4csbhypf 3439 . . . . . . . . 9  |-  ( y  =  A  ->  [_ y  /  x ]_ C  =  D )
65ineq1d 3685 . . . . . . . 8  |-  ( y  =  A  ->  ( [_ y  /  x ]_ C  i^i  [_ z  /  x ]_ C )  =  ( D  i^i  [_ z  /  x ]_ C ) )
76eqeq1d 2456 . . . . . . 7  |-  ( y  =  A  ->  (
( [_ y  /  x ]_ C  i^i  [_ z  /  x ]_ C )  =  (/)  <->  ( D  i^i  [_ z  /  x ]_ C )  =  (/) ) )
81, 7orbi12d 707 . . . . . 6  |-  ( y  =  A  ->  (
( y  =  z  \/  ( [_ y  /  x ]_ C  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<->  ( A  =  z  \/  ( D  i^i  [_ z  /  x ]_ C )  =  (/) ) ) )
98ralbidv 2893 . . . . 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 2458 . . . . . . 7  |-  ( y  =  B  ->  (
y  =  z  <->  B  =  z ) )
11 nfcv 2616 . . . . . . . . . 10  |-  F/_ x B
12 nfcv 2616 . . . . . . . . . 10  |-  F/_ x E
13 disjprg.2 . . . . . . . . . 10  |-  ( x  =  B  ->  C  =  E )
1411, 12, 13csbhypf 3439 . . . . . . . . 9  |-  ( y  =  B  ->  [_ y  /  x ]_ C  =  E )
1514ineq1d 3685 . . . . . . . 8  |-  ( y  =  B  ->  ( [_ y  /  x ]_ C  i^i  [_ z  /  x ]_ C )  =  ( E  i^i  [_ z  /  x ]_ C ) )
1615eqeq1d 2456 . . . . . . 7  |-  ( y  =  B  ->  (
( [_ y  /  x ]_ C  i^i  [_ z  /  x ]_ C )  =  (/)  <->  ( E  i^i  [_ z  /  x ]_ C )  =  (/) ) )
1710, 16orbi12d 707 . . . . . 6  |-  ( y  =  B  ->  (
( y  =  z  \/  ( [_ y  /  x ]_ C  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<->  ( B  =  z  \/  ( E  i^i  [_ z  /  x ]_ C )  =  (/) ) ) )
1817ralbidv 2893 . . . . 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 4065 . . . 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 1014 . . 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 22 . . . . . . . . . 10  |-  ( z  =  A  ->  z  =  A )
2221eqcomd 2462 . . . . . . . . 9  |-  ( z  =  A  ->  A  =  z )
2322orcd 390 . . . . . . . 8  |-  ( z  =  A  ->  ( A  =  z  \/  ( D  i^i  [_ z  /  x ]_ C )  =  (/) ) )
24 a1tru 1414 . . . . . . . 8  |-  ( z  =  A  -> T.  )
2523, 242thd 240 . . . . . . 7  |-  ( z  =  A  ->  (
( A  =  z  \/  ( D  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<-> T.  ) )
26 eqeq2 2469 . . . . . . . 8  |-  ( z  =  B  ->  ( A  =  z  <->  A  =  B ) )
2711, 12, 13csbhypf 3439 . . . . . . . . . 10  |-  ( z  =  B  ->  [_ z  /  x ]_ C  =  E )
2827ineq2d 3686 . . . . . . . . 9  |-  ( z  =  B  ->  ( D  i^i  [_ z  /  x ]_ C )  =  ( D  i^i  E ) )
2928eqeq1d 2456 . . . . . . . 8  |-  ( z  =  B  ->  (
( D  i^i  [_ z  /  x ]_ C )  =  (/)  <->  ( D  i^i  E )  =  (/) ) )
3026, 29orbi12d 707 . . . . . . 7  |-  ( z  =  B  ->  (
( A  =  z  \/  ( D  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<->  ( A  =  B  \/  ( D  i^i  E )  =  (/) ) ) )
3125, 30ralprg 4065 . . . . . 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 1014 . . . . 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 996 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B )  ->  A  =/=  B )
3433neneqd 2656 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B )  ->  -.  A  =  B
)
35 biorf 403 . . . . . . 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 1402 . . . . . . 7  |- T.
3837biantrur 504 . . . . . 6  |-  ( ( A  =  B  \/  ( D  i^i  E )  =  (/) )  <->  ( T.  /\  ( A  =  B  \/  ( D  i^i  E )  =  (/) ) ) )
3936, 38syl6bb 261 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B )  -> 
( ( D  i^i  E )  =  (/)  <->  ( T.  /\  ( A  =  B  \/  ( D  i^i  E )  =  (/) ) ) ) )
4032, 39bitr4d 256 . . . 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 2469 . . . . . . . . 9  |-  ( z  =  A  ->  ( B  =  z  <->  B  =  A ) )
42 eqcom 2463 . . . . . . . . 9  |-  ( B  =  A  <->  A  =  B )
4341, 42syl6bb 261 . . . . . . . 8  |-  ( z  =  A  ->  ( B  =  z  <->  A  =  B ) )
442, 3, 4csbhypf 3439 . . . . . . . . . . 11  |-  ( z  =  A  ->  [_ z  /  x ]_ C  =  D )
4544ineq2d 3686 . . . . . . . . . 10  |-  ( z  =  A  ->  ( E  i^i  [_ z  /  x ]_ C )  =  ( E  i^i  D ) )
46 incom 3677 . . . . . . . . . 10  |-  ( E  i^i  D )  =  ( D  i^i  E
)
4745, 46syl6eq 2511 . . . . . . . . 9  |-  ( z  =  A  ->  ( E  i^i  [_ z  /  x ]_ C )  =  ( D  i^i  E ) )
4847eqeq1d 2456 . . . . . . . 8  |-  ( z  =  A  ->  (
( E  i^i  [_ z  /  x ]_ C )  =  (/)  <->  ( D  i^i  E )  =  (/) ) )
4943, 48orbi12d 707 . . . . . . 7  |-  ( z  =  A  ->  (
( B  =  z  \/  ( E  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<->  ( A  =  B  \/  ( D  i^i  E )  =  (/) ) ) )
50 id 22 . . . . . . . . . 10  |-  ( z  =  B  ->  z  =  B )
5150eqcomd 2462 . . . . . . . . 9  |-  ( z  =  B  ->  B  =  z )
5251orcd 390 . . . . . . . 8  |-  ( z  =  B  ->  ( B  =  z  \/  ( E  i^i  [_ z  /  x ]_ C )  =  (/) ) )
53 a1tru 1414 . . . . . . . 8  |-  ( z  =  B  -> T.  )
5452, 532thd 240 . . . . . . 7  |-  ( z  =  B  ->  (
( B  =  z  \/  ( E  i^i  [_ z  /  x ]_ C )  =  (/) ) 
<-> T.  ) )
5549, 54ralprg 4065 . . . . . 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 1014 . . . . 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 503 . . . . . 6  |-  ( ( A  =  B  \/  ( D  i^i  E )  =  (/) )  <->  ( ( A  =  B  \/  ( D  i^i  E )  =  (/) )  /\ T.  ) )
5836, 57syl6bb 261 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B )  -> 
( ( D  i^i  E )  =  (/)  <->  ( ( A  =  B  \/  ( D  i^i  E )  =  (/) )  /\ T.  ) ) )
5956, 58bitr4d 256 . . . 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 708 . . 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 253 . 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 4425 . 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 pm4.24 641 . 2  |-  ( ( D  i^i  E )  =  (/)  <->  ( ( D  i^i  E )  =  (/)  /\  ( D  i^i  E )  =  (/) ) )
6461, 62, 633bitr4g 288 1  |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B )  -> 
(Disj  x  e.  { A ,  B } C  <->  ( D  i^i  E )  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1398   T. wtru 1399    e. wcel 1823    =/= wne 2649   A.wral 2804   [_csb 3420    i^i cin 3460   (/)c0 3783   {cpr 4018  Disj wdisj 4410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-nul 3784  df-sn 4017  df-pr 4019  df-disj 4411
This theorem is referenced by:  disjdifprg  27649  probun  28625
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