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Theorem txindis 20698
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 3754 . . . . . . 7  |-  ( -.  x  =  (/)  <->  E. y 
y  e.  x )
2 indistop 20066 . . . . . . . . . . 11  |-  { (/) ,  A }  e.  Top
3 indistop 20066 . . . . . . . . . . 11  |-  { (/) ,  B }  e.  Top
4 eltx 20632 . . . . . . . . . . 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 683 . . . . . . . . . 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 2766 . . . . . . . . . 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 200 . . . . . . . . 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 4241 . . . . . . . . . . . . . 14  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  ->  x  C_  U. ( {
(/) ,  A }  tX  { (/) ,  B }
) )
9 indisuni 20067 . . . . . . . . . . . . . . 15  |-  (  _I 
`  A )  = 
U. { (/) ,  A }
10 indisuni 20067 . . . . . . . . . . . . . . 15  |-  (  _I 
`  B )  = 
U. { (/) ,  B }
112, 3, 9, 10txunii 20657 . . . . . . . . . . . . . 14  |-  ( (  _I  `  A )  X.  (  _I  `  B ) )  = 
U. ( { (/) ,  A }  tX  { (/) ,  B } )
128, 11syl6sseqr 3491 . . . . . . . . . . . . 13  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  ->  x  C_  ( (  _I 
`  A )  X.  (  _I  `  B
) ) )
1312ad2antrr 737 . . . . . . . . . . . 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 3749 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  ( z  X.  w )  ->  (
z  X.  w )  =/=  (/) )
1514ad2antrl 739 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( z  X.  w )  =/=  (/) )
16 xpnz 5275 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  =/=  (/)  /\  w  =/=  (/) )  <->  ( z  X.  w )  =/=  (/) )
1715, 16sylibr 217 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( z  =/=  (/)  /\  w  =/=  (/) ) )
1817simpld 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  z  =/=  (/) )
1918neneqd 2640 . . . . . . . . . . . . . . . 16  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  -.  z  =  (/) )
20 simpll 765 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  z  e.  {
(/) ,  A }
)
21 indislem 20064 . . . . . . . . . . . . . . . . . . 19  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
2220, 21syl6eleqr 2551 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  z  e.  {
(/) ,  (  _I  `  A ) } )
23 elpri 3997 . . . . . . . . . . . . . . . . . 18  |-  ( z  e.  { (/) ,  (  _I  `  A ) }  ->  ( z  =  (/)  \/  z  =  (  _I  `  A
) ) )
2422, 23syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( z  =  (/)  \/  z  =  (  _I  `  A
) ) )
2524ord 383 . . . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . . . . . 17  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  w  =/=  (/) )
2827neneqd 2640 . . . . . . . . . . . . . . . 16  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  -.  w  =  (/) )
29 simplr 767 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  w  e.  {
(/) ,  B }
)
30 indislem 20064 . . . . . . . . . . . . . . . . . . 19  |-  { (/) ,  (  _I  `  B
) }  =  { (/)
,  B }
3129, 30syl6eleqr 2551 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  w  e.  {
(/) ,  (  _I  `  B ) } )
32 elpri 3997 . . . . . . . . . . . . . . . . . 18  |-  ( w  e.  { (/) ,  (  _I  `  B ) }  ->  ( w  =  (/)  \/  w  =  (  _I  `  B
) ) )
3331, 32syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( w  =  (/)  \/  w  =  (  _I  `  B
) ) )
3433ord 383 . . . . . . . . . . . . . . . 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 4878 . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . 14  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( z  X.  w )  C_  x
)
3836, 37eqsstr3d 3479 . . . . . . . . . . . . 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 725 . . . . . . . . . . . 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 3461 . . . . . . . . . . 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 440 . . . . . . . . . 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 2898 . . . . . . . . 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 45 . . . . . . . 8  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( y  e.  x  ->  x  =  ( (  _I  `  A )  X.  (  _I  `  B ) ) ) )
4443exlimdv 1790 . . . . . . 7  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( E. y  y  e.  x  ->  x  =  ( (  _I 
`  A )  X.  (  _I  `  B
) ) ) )
451, 44syl5bi 225 . . . . . 6  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( -.  x  =  (/)  ->  x  =  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) ) )
4645orrd 384 . . . . 5  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( x  =  (/)  \/  x  =  ( (  _I  `  A )  X.  (  _I  `  B ) ) ) )
47 vex 3060 . . . . . 6  |-  x  e. 
_V
4847elpr 3998 . . . . 5  |-  ( x  e.  { (/) ,  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) }  <->  ( x  =  (/)  \/  x  =  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) ) )
4946, 48sylibr 217 . . . 4  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  ->  x  e.  { (/) ,  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) } )
5049ssriv 3448 . . 3  |-  ( {
(/) ,  A }  tX  { (/) ,  B }
)  C_  { (/) ,  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) }
519toptopon 19997 . . . . . . 7  |-  ( {
(/) ,  A }  e.  Top  <->  { (/) ,  A }  e.  (TopOn `  (  _I  `  A ) ) )
522, 51mpbi 213 . . . . . 6  |-  { (/) ,  A }  e.  (TopOn `  (  _I  `  A
) )
5310toptopon 19997 . . . . . . 7  |-  ( {
(/) ,  B }  e.  Top  <->  { (/) ,  B }  e.  (TopOn `  (  _I  `  B ) ) )
543, 53mpbi 213 . . . . . 6  |-  { (/) ,  B }  e.  (TopOn `  (  _I  `  B
) )
55 txtopon 20655 . . . . . 6  |-  ( ( { (/) ,  A }  e.  (TopOn `  (  _I  `  A ) )  /\  {
(/) ,  B }  e.  (TopOn `  (  _I  `  B ) ) )  ->  ( { (/) ,  A }  tX  { (/) ,  B } )  e.  (TopOn `  ( (  _I  `  A )  X.  (  _I  `  B
) ) ) )
5652, 54, 55mp2an 683 . . . . 5  |-  ( {
(/) ,  A }  tX  { (/) ,  B }
)  e.  (TopOn `  ( (  _I  `  A )  X.  (  _I  `  B ) ) )
57 topgele 19998 . . . . 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 5 . . . 4  |-  ( {
(/) ,  ( (  _I  `  A )  X.  (  _I  `  B
) ) }  C_  ( { (/) ,  A }  tX  { (/) ,  B }
)  /\  ( { (/)
,  A }  tX  {
(/) ,  B }
)  C_  ~P (
(  _I  `  A
)  X.  (  _I 
`  B ) ) )
5958simpli 464 . . 3  |-  { (/) ,  ( (  _I  `  A )  X.  (  _I  `  B ) ) }  C_  ( { (/)
,  A }  tX  {
(/) ,  B }
)
6050, 59eqssi 3460 . 2  |-  ( {
(/) ,  A }  tX  { (/) ,  B }
)  =  { (/) ,  ( (  _I  `  A )  X.  (  _I  `  B ) ) }
61 txindislem 20697 . . 3  |-  ( (  _I  `  A )  X.  (  _I  `  B ) )  =  (  _I  `  ( A  X.  B ) )
6261preq2i 4068 . 2  |-  { (/) ,  ( (  _I  `  A )  X.  (  _I  `  B ) ) }  =  { (/) ,  (  _I  `  ( A  X.  B ) ) }
63 indislem 20064 . 2  |-  { (/) ,  (  _I  `  ( A  X.  B ) ) }  =  { (/) ,  ( A  X.  B
) }
6460, 62, 633eqtri 2488 1  |-  ( {
(/) ,  A }  tX  { (/) ,  B }
)  =  { (/) ,  ( A  X.  B
) }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375    = wceq 1455   E.wex 1674    e. wcel 1898    =/= wne 2633   A.wral 2749   E.wrex 2750    C_ wss 3416   (/)c0 3743   ~Pcpw 3963   {cpr 3982   U.cuni 4212    _I cid 4763    X. cxp 4851   ` cfv 5601  (class class class)co 6315   Topctop 19966  TopOnctopon 19967    tX ctx 20624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-1st 6820  df-2nd 6821  df-topgen 15391  df-top 19970  df-bases 19971  df-topon 19972  df-tx 20626
This theorem is referenced by: (None)
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