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Theorem txindis 19107
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 3644 . . . . . . 7  |-  ( -.  x  =  (/)  <->  E. y 
y  e.  x )
2 indistop 18506 . . . . . . . . . . 11  |-  { (/) ,  A }  e.  Top
3 indistop 18506 . . . . . . . . . . 11  |-  { (/) ,  B }  e.  Top
4 eltx 19041 . . . . . . . . . . 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 667 . . . . . . . . . 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 2774 . . . . . . . . . 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 195 . . . . . . . . 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 4118 . . . . . . . . . . . . . 14  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  ->  x  C_  U. ( {
(/) ,  A }  tX  { (/) ,  B }
) )
9 indisuni 18507 . . . . . . . . . . . . . . 15  |-  (  _I 
`  A )  = 
U. { (/) ,  A }
10 indisuni 18507 . . . . . . . . . . . . . . 15  |-  (  _I 
`  B )  = 
U. { (/) ,  B }
112, 3, 9, 10txunii 19066 . . . . . . . . . . . . . 14  |-  ( (  _I  `  A )  X.  (  _I  `  B ) )  = 
U. ( { (/) ,  A }  tX  { (/) ,  B } )
128, 11syl6sseqr 3400 . . . . . . . . . . . . 13  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  ->  x  C_  ( (  _I 
`  A )  X.  (  _I  `  B
) ) )
1312ad2antrr 720 . . . . . . . . . . . 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 3640 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  ( z  X.  w )  ->  (
z  X.  w )  =/=  (/) )
1514ad2antrl 722 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( z  X.  w )  =/=  (/) )
16 xpnz 5254 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  =/=  (/)  /\  w  =/=  (/) )  <->  ( z  X.  w )  =/=  (/) )
1715, 16sylibr 212 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( z  =/=  (/)  /\  w  =/=  (/) ) )
1817simpld 456 . . . . . . . . . . . . . . . . 17  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  z  =/=  (/) )
1918neneqd 2622 . . . . . . . . . . . . . . . 16  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  -.  z  =  (/) )
20 simpll 748 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  z  e.  {
(/) ,  A }
)
21 indislem 18504 . . . . . . . . . . . . . . . . . . 19  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
2220, 21syl6eleqr 2532 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  z  e.  {
(/) ,  (  _I  `  A ) } )
23 elpri 3894 . . . . . . . . . . . . . . . . . 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 377 . . . . . . . . . . . . . . . 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 460 . . . . . . . . . . . . . . . . 17  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  w  =/=  (/) )
2827neneqd 2622 . . . . . . . . . . . . . . . 16  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  -.  w  =  (/) )
29 simplr 749 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  w  e.  {
(/) ,  B }
)
30 indislem 18504 . . . . . . . . . . . . . . . . . . 19  |-  { (/) ,  (  _I  `  B
) }  =  { (/)
,  B }
3129, 30syl6eleqr 2532 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  w  e.  {
(/) ,  (  _I  `  B ) } )
32 elpri 3894 . . . . . . . . . . . . . . . . . 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 377 . . . . . . . . . . . . . . . 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 4861 . . . . . . . . . . . . . 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 751 . . . . . . . . . . . . . 14  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( z  X.  w )  C_  x
)
3836, 37eqsstr3d 3388 . . . . . . . . . . . . 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 708 . . . . . . . . . . . 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 3370 . . . . . . . . . . 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 434 . . . . . . . . . 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 2846 . . . . . . . . 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 44 . . . . . . . 8  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( y  e.  x  ->  x  =  ( (  _I  `  A )  X.  (  _I  `  B ) ) ) )
4443exlimdv 1695 . . . . . . 7  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( E. y  y  e.  x  ->  x  =  ( (  _I 
`  A )  X.  (  _I  `  B
) ) ) )
451, 44syl5bi 217 . . . . . 6  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( -.  x  =  (/)  ->  x  =  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) ) )
4645orrd 378 . . . . 5  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( x  =  (/)  \/  x  =  ( (  _I  `  A )  X.  (  _I  `  B ) ) ) )
47 vex 2973 . . . . . 6  |-  x  e. 
_V
4847elpr 3892 . . . . 5  |-  ( x  e.  { (/) ,  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) }  <->  ( x  =  (/)  \/  x  =  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) ) )
4946, 48sylibr 212 . . . 4  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  ->  x  e.  { (/) ,  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) } )
5049ssriv 3357 . . 3  |-  ( {
(/) ,  A }  tX  { (/) ,  B }
)  C_  { (/) ,  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) }
519toptopon 18438 . . . . . . 7  |-  ( {
(/) ,  A }  e.  Top  <->  { (/) ,  A }  e.  (TopOn `  (  _I  `  A ) ) )
522, 51mpbi 208 . . . . . 6  |-  { (/) ,  A }  e.  (TopOn `  (  _I  `  A
) )
5310toptopon 18438 . . . . . . 7  |-  ( {
(/) ,  B }  e.  Top  <->  { (/) ,  B }  e.  (TopOn `  (  _I  `  B ) ) )
543, 53mpbi 208 . . . . . 6  |-  { (/) ,  B }  e.  (TopOn `  (  _I  `  B
) )
55 txtopon 19064 . . . . . 6  |-  ( ( { (/) ,  A }  e.  (TopOn `  (  _I  `  A ) )  /\  {
(/) ,  B }  e.  (TopOn `  (  _I  `  B ) ) )  ->  ( { (/) ,  A }  tX  { (/) ,  B } )  e.  (TopOn `  ( (  _I  `  A )  X.  (  _I  `  B
) ) ) )
5652, 54, 55mp2an 667 . . . . 5  |-  ( {
(/) ,  A }  tX  { (/) ,  B }
)  e.  (TopOn `  ( (  _I  `  A )  X.  (  _I  `  B ) ) )
57 topgele 18439 . . . . 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 455 . . 3  |-  { (/) ,  ( (  _I  `  A )  X.  (  _I  `  B ) ) }  C_  ( { (/)
,  A }  tX  {
(/) ,  B }
)
6050, 59eqssi 3369 . 2  |-  ( {
(/) ,  A }  tX  { (/) ,  B }
)  =  { (/) ,  ( (  _I  `  A )  X.  (  _I  `  B ) ) }
61 txindislem 19106 . . 3  |-  ( (  _I  `  A )  X.  (  _I  `  B ) )  =  (  _I  `  ( A  X.  B ) )
6261preq2i 3955 . 2  |-  { (/) ,  ( (  _I  `  A )  X.  (  _I  `  B ) ) }  =  { (/) ,  (  _I  `  ( A  X.  B ) ) }
63 indislem 18504 . 2  |-  { (/) ,  (  _I  `  ( A  X.  B ) ) }  =  { (/) ,  ( A  X.  B
) }
6460, 62, 633eqtri 2465 1  |-  ( {
(/) ,  A }  tX  { (/) ,  B }
)  =  { (/) ,  ( A  X.  B
) }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1364   E.wex 1591    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714    C_ wss 3325   (/)c0 3634   ~Pcpw 3857   {cpr 3876   U.cuni 4088    _I cid 4627    X. cxp 4834   ` cfv 5415  (class class class)co 6090   Topctop 18398  TopOnctopon 18399    tX ctx 19033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-topgen 14378  df-top 18403  df-bases 18405  df-topon 18406  df-tx 19035
This theorem is referenced by: (None)
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