Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjdifprg Structured version   Unicode version

Theorem disjdifprg 27867
Description: A trivial partition into a subset and its complement. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
disjdifprg  |-  ( ( A  e.  V  /\  B  e.  W )  -> Disj  x  e.  { ( B  \  A ) ,  A } x )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    V( x)    W( x)

Proof of Theorem disjdifprg
StepHypRef Expression
1 disjxsn 4389 . . . . . 6  |- Disj  x  e. 
{ (/) } x
2 simpr 459 . . . . . . . 8  |-  ( ( B  e.  W  /\  B  =  (/) )  ->  B  =  (/) )
3 eqidd 2403 . . . . . . . 8  |-  ( ( B  e.  W  /\  B  =  (/) )  ->  (/)  =  (/) )
4 elex 3068 . . . . . . . . . 10  |-  ( B  e.  W  ->  B  e.  _V )
5 0ex 4526 . . . . . . . . . . 11  |-  (/)  e.  _V
65a1i 11 . . . . . . . . . 10  |-  ( B  e.  W  ->  (/)  e.  _V )
74, 6, 6preqsnd 27839 . . . . . . . . 9  |-  ( B  e.  W  ->  ( { B ,  (/) }  =  { (/) }  <->  ( B  =  (/)  /\  (/)  =  (/) ) ) )
87adantr 463 . . . . . . . 8  |-  ( ( B  e.  W  /\  B  =  (/) )  -> 
( { B ,  (/)
}  =  { (/) }  <-> 
( B  =  (/)  /\  (/)  =  (/) ) ) )
92, 3, 8mpbir2and 923 . . . . . . 7  |-  ( ( B  e.  W  /\  B  =  (/) )  ->  { B ,  (/) }  =  { (/) } )
109disjeq1d 4374 . . . . . 6  |-  ( ( B  e.  W  /\  B  =  (/) )  -> 
(Disj  x  e.  { B ,  (/) } x  <-> Disj  x  e.  { (/)
} x ) )
111, 10mpbiri 233 . . . . 5  |-  ( ( B  e.  W  /\  B  =  (/) )  -> Disj  x  e.  { B ,  (/)
} x )
12 in0 3765 . . . . . 6  |-  ( B  i^i  (/) )  =  (/)
134adantr 463 . . . . . . 7  |-  ( ( B  e.  W  /\  B  =/=  (/) )  ->  B  e.  _V )
145a1i 11 . . . . . . 7  |-  ( ( B  e.  W  /\  B  =/=  (/) )  ->  (/)  e.  _V )
15 simpr 459 . . . . . . 7  |-  ( ( B  e.  W  /\  B  =/=  (/) )  ->  B  =/=  (/) )
16 id 22 . . . . . . . 8  |-  ( x  =  B  ->  x  =  B )
17 id 22 . . . . . . . 8  |-  ( x  =  (/)  ->  x  =  (/) )
1816, 17disjprg 4391 . . . . . . 7  |-  ( ( B  e.  _V  /\  (/) 
e.  _V  /\  B  =/=  (/) )  ->  (Disj  x  e.  { B ,  (/) } x  <->  ( B  i^i  (/) )  =  (/) ) )
1913, 14, 15, 18syl3anc 1230 . . . . . 6  |-  ( ( B  e.  W  /\  B  =/=  (/) )  ->  (Disj  x  e.  { B ,  (/)
} x  <->  ( B  i^i  (/) )  =  (/) ) )
2012, 19mpbiri 233 . . . . 5  |-  ( ( B  e.  W  /\  B  =/=  (/) )  -> Disj  x  e. 
{ B ,  (/) } x )
2111, 20pm2.61dane 2721 . . . 4  |-  ( B  e.  W  -> Disj  x  e. 
{ B ,  (/) } x )
2221ad2antlr 725 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  A  =  (/) )  -> Disj  x  e.  { B ,  (/) } x
)
23 difeq2 3555 . . . . . . 7  |-  ( A  =  (/)  ->  ( B 
\  A )  =  ( B  \  (/) ) )
24 dif0 3842 . . . . . . 7  |-  ( B 
\  (/) )  =  B
2523, 24syl6eq 2459 . . . . . 6  |-  ( A  =  (/)  ->  ( B 
\  A )  =  B )
26 id 22 . . . . . 6  |-  ( A  =  (/)  ->  A  =  (/) )
2725, 26preq12d 4059 . . . . 5  |-  ( A  =  (/)  ->  { ( B  \  A ) ,  A }  =  { B ,  (/) } )
2827disjeq1d 4374 . . . 4  |-  ( A  =  (/)  ->  (Disj  x  e.  { ( B  \  A ) ,  A } x  <-> Disj  x  e.  { B ,  (/) } x ) )
2928adantl 464 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  A  =  (/) )  ->  (Disj  x  e. 
{ ( B  \  A ) ,  A } x  <-> Disj  x  e.  { B ,  (/) } x ) )
3022, 29mpbird 232 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  A  =  (/) )  -> Disj  x  e.  {
( B  \  A
) ,  A }
x )
31 incom 3632 . . . 4  |-  ( A  i^i  ( B  \  A ) )  =  ( ( B  \  A )  i^i  A
)
32 disjdif 3844 . . . 4  |-  ( A  i^i  ( B  \  A ) )  =  (/)
3331, 32eqtr3i 2433 . . 3  |-  ( ( B  \  A )  i^i  A )  =  (/)
34 difexg 4542 . . . . 5  |-  ( B  e.  W  ->  ( B  \  A )  e. 
_V )
3534ad2antlr 725 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  -.  A  =  (/) )  ->  ( B  \  A )  e. 
_V )
36 elex 3068 . . . . 5  |-  ( A  e.  V  ->  A  e.  _V )
3736ad2antrr 724 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  -.  A  =  (/) )  ->  A  e.  _V )
38 ssid 3461 . . . . . 6  |-  ( B 
\  A )  C_  ( B  \  A )
39 ssdifeq0 3854 . . . . . . . 8  |-  ( A 
C_  ( B  \  A )  <->  A  =  (/) )
4039notbii 294 . . . . . . 7  |-  ( -.  A  C_  ( B  \  A )  <->  -.  A  =  (/) )
41 nssne2 3499 . . . . . . 7  |-  ( ( ( B  \  A
)  C_  ( B  \  A )  /\  -.  A  C_  ( B  \  A ) )  -> 
( B  \  A
)  =/=  A )
4240, 41sylan2br 474 . . . . . 6  |-  ( ( ( B  \  A
)  C_  ( B  \  A )  /\  -.  A  =  (/) )  -> 
( B  \  A
)  =/=  A )
4338, 42mpan 668 . . . . 5  |-  ( -.  A  =  (/)  ->  ( B  \  A )  =/= 
A )
4443adantl 464 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  -.  A  =  (/) )  ->  ( B  \  A )  =/= 
A )
45 id 22 . . . . 5  |-  ( x  =  ( B  \  A )  ->  x  =  ( B  \  A ) )
46 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
4745, 46disjprg 4391 . . . 4  |-  ( ( ( B  \  A
)  e.  _V  /\  A  e.  _V  /\  ( B  \  A )  =/= 
A )  ->  (Disj  x  e.  { ( B 
\  A ) ,  A } x  <->  ( ( B  \  A )  i^i 
A )  =  (/) ) )
4835, 37, 44, 47syl3anc 1230 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  -.  A  =  (/) )  ->  (Disj  x  e.  { ( B 
\  A ) ,  A } x  <->  ( ( B  \  A )  i^i 
A )  =  (/) ) )
4933, 48mpbiri 233 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  -.  A  =  (/) )  -> Disj  x  e. 
{ ( B  \  A ) ,  A } x )
5030, 49pm2.61dan 792 1  |-  ( ( A  e.  V  /\  B  e.  W )  -> Disj  x  e.  { ( B  \  A ) ,  A } x )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   _Vcvv 3059    \ cdif 3411    i^i cin 3413    C_ wss 3414   (/)c0 3738   {csn 3972   {cpr 3974  Disj wdisj 4366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-sn 3973  df-pr 3975  df-disj 4367
This theorem is referenced by:  disjdifprg2  27868  measssd  28663
  Copyright terms: Public domain W3C validator