Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > txdis | Unicode version |
Description: The topological product of discrete spaces is discrete. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
txdis |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | distop 12626 | . . . . 5 | |
2 | distop 12626 | . . . . 5 | |
3 | unipw 4189 | . . . . . . 7 | |
4 | 3 | eqcomi 2168 | . . . . . 6 |
5 | unipw 4189 | . . . . . . 7 | |
6 | 5 | eqcomi 2168 | . . . . . 6 |
7 | 4, 6 | txuni 12804 | . . . . 5 |
8 | 1, 2, 7 | syl2an 287 | . . . 4 |
9 | eqimss2 3192 | . . . 4 | |
10 | 8, 9 | syl 14 | . . 3 |
11 | sspwuni 3944 | . . 3 | |
12 | 10, 11 | sylibr 133 | . 2 |
13 | elelpwi 3565 | . . . . . . . . 9 | |
14 | 13 | adantl 275 | . . . . . . . 8 |
15 | xp1st 6125 | . . . . . . . 8 | |
16 | snelpwi 4184 | . . . . . . . 8 | |
17 | 14, 15, 16 | 3syl 17 | . . . . . . 7 |
18 | xp2nd 6126 | . . . . . . . 8 | |
19 | snelpwi 4184 | . . . . . . . 8 | |
20 | 14, 18, 19 | 3syl 17 | . . . . . . 7 |
21 | vsnid 3602 | . . . . . . . 8 | |
22 | 1st2nd2 6135 | . . . . . . . . . 10 | |
23 | 14, 22 | syl 14 | . . . . . . . . 9 |
24 | 23 | sneqd 3583 | . . . . . . . 8 |
25 | 21, 24 | eleqtrid 2253 | . . . . . . 7 |
26 | simprl 521 | . . . . . . . . 9 | |
27 | 23, 26 | eqeltrrd 2242 | . . . . . . . 8 |
28 | 27 | snssd 3712 | . . . . . . 7 |
29 | xpeq1 4612 | . . . . . . . . . 10 | |
30 | 29 | eleq2d 2234 | . . . . . . . . 9 |
31 | 29 | sseq1d 3166 | . . . . . . . . 9 |
32 | 30, 31 | anbi12d 465 | . . . . . . . 8 |
33 | xpeq2 4613 | . . . . . . . . . . 11 | |
34 | 1stexg 6127 | . . . . . . . . . . . . 13 | |
35 | 34 | elv 2725 | . . . . . . . . . . . 12 |
36 | 2ndexg 6128 | . . . . . . . . . . . . 13 | |
37 | 36 | elv 2725 | . . . . . . . . . . . 12 |
38 | 35, 37 | xpsn 5655 | . . . . . . . . . . 11 |
39 | 33, 38 | eqtrdi 2213 | . . . . . . . . . 10 |
40 | 39 | eleq2d 2234 | . . . . . . . . 9 |
41 | 39 | sseq1d 3166 | . . . . . . . . 9 |
42 | 40, 41 | anbi12d 465 | . . . . . . . 8 |
43 | 32, 42 | rspc2ev 2840 | . . . . . . 7 |
44 | 17, 20, 25, 28, 43 | syl112anc 1231 | . . . . . 6 |
45 | 44 | expr 373 | . . . . 5 |
46 | 45 | ralrimdva 2544 | . . . 4 |
47 | eltx 12800 | . . . . 5 | |
48 | 1, 2, 47 | syl2an 287 | . . . 4 |
49 | 46, 48 | sylibrd 168 | . . 3 |
50 | 49 | ssrdv 3143 | . 2 |
51 | 12, 50 | eqssd 3154 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wral 2442 wrex 2443 cvv 2721 wss 3111 cpw 3553 csn 3570 cop 3573 cuni 3783 cxp 4596 cfv 5182 (class class class)co 5836 c1st 6098 c2nd 6099 ctop 12536 ctx 12793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-topgen 12513 df-top 12537 df-topon 12550 df-bases 12582 df-tx 12794 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |