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Theorem txindis 17667
Description: The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
txindis  |-  ( {
(/) ,  A }  tX  { (/) ,  B }
)  =  { (/) ,  ( A  X.  B
) }

Proof of Theorem txindis
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neq0 3639 . . . . . . 7  |-  ( -.  x  =  (/)  <->  E. y 
y  e.  x )
2 indistop 17067 . . . . . . . . . . 11  |-  { (/) ,  A }  e.  Top
3 indistop 17067 . . . . . . . . . . 11  |-  { (/) ,  B }  e.  Top
4 eltx 17601 . . . . . . . . . . 11  |-  ( ( { (/) ,  A }  e.  Top  /\  { (/) ,  B }  e.  Top )  ->  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  <->  A. y  e.  x  E. z  e.  { (/) ,  A } E. w  e.  { (/) ,  B }  ( y  e.  ( z  X.  w )  /\  (
z  X.  w ) 
C_  x ) ) )
52, 3, 4mp2an 655 . . . . . . . . . 10  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  <->  A. y  e.  x  E. z  e.  { (/) ,  A } E. w  e.  { (/) ,  B }  ( y  e.  ( z  X.  w )  /\  (
z  X.  w ) 
C_  x ) )
6 rsp 2767 . . . . . . . . . 10  |-  ( A. y  e.  x  E. z  e.  { (/) ,  A } E. w  e.  { (/)
,  B }  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )  ->  ( y  e.  x  ->  E. z  e.  { (/)
,  A } E. w  e.  { (/) ,  B }  ( y  e.  ( z  X.  w
)  /\  ( z  X.  w )  C_  x
) ) )
75, 6sylbi 189 . . . . . . . . 9  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( y  e.  x  ->  E. z  e.  { (/)
,  A } E. w  e.  { (/) ,  B }  ( y  e.  ( z  X.  w
)  /\  ( z  X.  w )  C_  x
) ) )
8 elssuni 4044 . . . . . . . . . . . . . 14  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  ->  x  C_  U. ( {
(/) ,  A }  tX  { (/) ,  B }
) )
9 indisuni 17068 . . . . . . . . . . . . . . 15  |-  (  _I 
`  A )  = 
U. { (/) ,  A }
10 indisuni 17068 . . . . . . . . . . . . . . 15  |-  (  _I 
`  B )  = 
U. { (/) ,  B }
112, 3, 9, 10txunii 17626 . . . . . . . . . . . . . 14  |-  ( (  _I  `  A )  X.  (  _I  `  B ) )  = 
U. ( { (/) ,  A }  tX  { (/) ,  B } )
128, 11syl6sseqr 3396 . . . . . . . . . . . . 13  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  ->  x  C_  ( (  _I 
`  A )  X.  (  _I  `  B
) ) )
1312ad2antrr 708 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B }
)  /\  ( z  e.  { (/) ,  A }  /\  w  e.  { (/) ,  B } ) )  /\  ( y  e.  ( z  X.  w
)  /\  ( z  X.  w )  C_  x
) )  ->  x  C_  ( (  _I  `  A )  X.  (  _I  `  B ) ) )
14 ne0i 3635 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  ( z  X.  w )  ->  (
z  X.  w )  =/=  (/) )
1514ad2antrl 710 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( z  X.  w )  =/=  (/) )
16 xpnz 5293 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  =/=  (/)  /\  w  =/=  (/) )  <->  ( z  X.  w )  =/=  (/) )
1715, 16sylibr 205 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( z  =/=  (/)  /\  w  =/=  (/) ) )
1817simpld 447 . . . . . . . . . . . . . . . . 17  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  z  =/=  (/) )
1918neneqd 2618 . . . . . . . . . . . . . . . 16  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  -.  z  =  (/) )
20 simpll 732 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  z  e.  {
(/) ,  A }
)
21 indislem 17065 . . . . . . . . . . . . . . . . . . 19  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
2220, 21syl6eleqr 2528 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  z  e.  {
(/) ,  (  _I  `  A ) } )
23 elpri 3835 . . . . . . . . . . . . . . . . . 18  |-  ( z  e.  { (/) ,  (  _I  `  A ) }  ->  ( z  =  (/)  \/  z  =  (  _I  `  A
) ) )
2422, 23syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( z  =  (/)  \/  z  =  (  _I  `  A
) ) )
2524ord 368 . . . . . . . . . . . . . . . 16  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( -.  z  =  (/)  ->  z  =  (  _I  `  A
) ) )
2619, 25mpd 15 . . . . . . . . . . . . . . 15  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  z  =  (  _I  `  A ) )
2717simprd 451 . . . . . . . . . . . . . . . . 17  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  w  =/=  (/) )
2827neneqd 2618 . . . . . . . . . . . . . . . 16  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  -.  w  =  (/) )
29 simplr 733 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  w  e.  {
(/) ,  B }
)
30 indislem 17065 . . . . . . . . . . . . . . . . . . 19  |-  { (/) ,  (  _I  `  B
) }  =  { (/)
,  B }
3129, 30syl6eleqr 2528 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  w  e.  {
(/) ,  (  _I  `  B ) } )
32 elpri 3835 . . . . . . . . . . . . . . . . . 18  |-  ( w  e.  { (/) ,  (  _I  `  B ) }  ->  ( w  =  (/)  \/  w  =  (  _I  `  B
) ) )
3331, 32syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( w  =  (/)  \/  w  =  (  _I  `  B
) ) )
3433ord 368 . . . . . . . . . . . . . . . 16  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( -.  w  =  (/)  ->  w  =  (  _I  `  B
) ) )
3528, 34mpd 15 . . . . . . . . . . . . . . 15  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  w  =  (  _I  `  B ) )
3626, 35xpeq12d 4904 . . . . . . . . . . . . . 14  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( z  X.  w )  =  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) )
37 simprr 735 . . . . . . . . . . . . . 14  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( z  X.  w )  C_  x
)
3836, 37eqsstr3d 3384 . . . . . . . . . . . . 13  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( (  _I  `  A )  X.  (  _I  `  B
) )  C_  x
)
3938adantll 696 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B }
)  /\  ( z  e.  { (/) ,  A }  /\  w  e.  { (/) ,  B } ) )  /\  ( y  e.  ( z  X.  w
)  /\  ( z  X.  w )  C_  x
) )  ->  (
(  _I  `  A
)  X.  (  _I 
`  B ) ) 
C_  x )
4013, 39eqssd 3366 . . . . . . . . . . 11  |-  ( ( ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B }
)  /\  ( z  e.  { (/) ,  A }  /\  w  e.  { (/) ,  B } ) )  /\  ( y  e.  ( z  X.  w
)  /\  ( z  X.  w )  C_  x
) )  ->  x  =  ( (  _I 
`  A )  X.  (  _I  `  B
) ) )
4140ex 425 . . . . . . . . . 10  |-  ( ( x  e.  ( {
(/) ,  A }  tX  { (/) ,  B }
)  /\  ( z  e.  { (/) ,  A }  /\  w  e.  { (/) ,  B } ) )  ->  ( ( y  e.  ( z  X.  w )  /\  (
z  X.  w ) 
C_  x )  ->  x  =  ( (  _I  `  A )  X.  (  _I  `  B
) ) ) )
4241rexlimdvva 2838 . . . . . . . . 9  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( E. z  e. 
{ (/) ,  A } E. w  e.  { (/) ,  B }  ( y  e.  ( z  X.  w )  /\  (
z  X.  w ) 
C_  x )  ->  x  =  ( (  _I  `  A )  X.  (  _I  `  B
) ) ) )
437, 42syld 43 . . . . . . . 8  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( y  e.  x  ->  x  =  ( (  _I  `  A )  X.  (  _I  `  B ) ) ) )
4443exlimdv 1647 . . . . . . 7  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( E. y  y  e.  x  ->  x  =  ( (  _I 
`  A )  X.  (  _I  `  B
) ) ) )
451, 44syl5bi 210 . . . . . 6  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( -.  x  =  (/)  ->  x  =  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) ) )
4645orrd 369 . . . . 5  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( x  =  (/)  \/  x  =  ( (  _I  `  A )  X.  (  _I  `  B ) ) ) )
47 vex 2960 . . . . . 6  |-  x  e. 
_V
4847elpr 3833 . . . . 5  |-  ( x  e.  { (/) ,  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) }  <->  ( x  =  (/)  \/  x  =  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) ) )
4946, 48sylibr 205 . . . 4  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  ->  x  e.  { (/) ,  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) } )
5049ssriv 3353 . . 3  |-  ( {
(/) ,  A }  tX  { (/) ,  B }
)  C_  { (/) ,  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) }
519toptopon 16999 . . . . . . 7  |-  ( {
(/) ,  A }  e.  Top  <->  { (/) ,  A }  e.  (TopOn `  (  _I  `  A ) ) )
522, 51mpbi 201 . . . . . 6  |-  { (/) ,  A }  e.  (TopOn `  (  _I  `  A
) )
5310toptopon 16999 . . . . . . 7  |-  ( {
(/) ,  B }  e.  Top  <->  { (/) ,  B }  e.  (TopOn `  (  _I  `  B ) ) )
543, 53mpbi 201 . . . . . 6  |-  { (/) ,  B }  e.  (TopOn `  (  _I  `  B
) )
55 txtopon 17624 . . . . . 6  |-  ( ( { (/) ,  A }  e.  (TopOn `  (  _I  `  A ) )  /\  {
(/) ,  B }  e.  (TopOn `  (  _I  `  B ) ) )  ->  ( { (/) ,  A }  tX  { (/) ,  B } )  e.  (TopOn `  ( (  _I  `  A )  X.  (  _I  `  B
) ) ) )
5652, 54, 55mp2an 655 . . . . 5  |-  ( {
(/) ,  A }  tX  { (/) ,  B }
)  e.  (TopOn `  ( (  _I  `  A )  X.  (  _I  `  B ) ) )
57 topgele 17000 . . . . 5  |-  ( ( { (/) ,  A }  tX  { (/) ,  B }
)  e.  (TopOn `  ( (  _I  `  A )  X.  (  _I  `  B ) ) )  ->  ( { (/)
,  ( (  _I 
`  A )  X.  (  _I  `  B
) ) }  C_  ( { (/) ,  A }  tX  { (/) ,  B }
)  /\  ( { (/)
,  A }  tX  {
(/) ,  B }
)  C_  ~P (
(  _I  `  A
)  X.  (  _I 
`  B ) ) ) )
5856, 57ax-mp 8 . . . 4  |-  ( {
(/) ,  ( (  _I  `  A )  X.  (  _I  `  B
) ) }  C_  ( { (/) ,  A }  tX  { (/) ,  B }
)  /\  ( { (/)
,  A }  tX  {
(/) ,  B }
)  C_  ~P (
(  _I  `  A
)  X.  (  _I 
`  B ) ) )
5958simpli 446 . . 3  |-  { (/) ,  ( (  _I  `  A )  X.  (  _I  `  B ) ) }  C_  ( { (/)
,  A }  tX  {
(/) ,  B }
)
6050, 59eqssi 3365 . 2  |-  ( {
(/) ,  A }  tX  { (/) ,  B }
)  =  { (/) ,  ( (  _I  `  A )  X.  (  _I  `  B ) ) }
61 txindislem 17666 . . 3  |-  ( (  _I  `  A )  X.  (  _I  `  B ) )  =  (  _I  `  ( A  X.  B ) )
6261preq2i 3888 . 2  |-  { (/) ,  ( (  _I  `  A )  X.  (  _I  `  B ) ) }  =  { (/) ,  (  _I  `  ( A  X.  B ) ) }
63 indislem 17065 . 2  |-  { (/) ,  (  _I  `  ( A  X.  B ) ) }  =  { (/) ,  ( A  X.  B
) }
6460, 62, 633eqtri 2461 1  |-  ( {
(/) ,  A }  tX  { (/) ,  B }
)  =  { (/) ,  ( A  X.  B
) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2600   A.wral 2706   E.wrex 2707    C_ wss 3321   (/)c0 3629   ~Pcpw 3800   {cpr 3816   U.cuni 4016    _I cid 4494    X. cxp 4877   ` cfv 5455  (class class class)co 6082   Topctop 16959  TopOnctopon 16960    tX ctx 17593
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-topgen 13668  df-top 16964  df-bases 16966  df-topon 16967  df-tx 17595
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