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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 13165 | . . . . 5 | |
2 | distop 13165 | . . . . 5 | |
3 | unipw 4211 | . . . . . . 7 | |
4 | 3 | eqcomi 2179 | . . . . . 6 |
5 | unipw 4211 | . . . . . . 7 | |
6 | 5 | eqcomi 2179 | . . . . . 6 |
7 | 4, 6 | txuni 13343 | . . . . 5 |
8 | 1, 2, 7 | syl2an 289 | . . . 4 |
9 | eqimss2 3208 | . . . 4 | |
10 | 8, 9 | syl 14 | . . 3 |
11 | sspwuni 3966 | . . 3 | |
12 | 10, 11 | sylibr 134 | . 2 |
13 | elelpwi 3584 | . . . . . . . . 9 | |
14 | 13 | adantl 277 | . . . . . . . 8 |
15 | xp1st 6156 | . . . . . . . 8 | |
16 | snelpwi 4206 | . . . . . . . 8 | |
17 | 14, 15, 16 | 3syl 17 | . . . . . . 7 |
18 | xp2nd 6157 | . . . . . . . 8 | |
19 | snelpwi 4206 | . . . . . . . 8 | |
20 | 14, 18, 19 | 3syl 17 | . . . . . . 7 |
21 | vsnid 3621 | . . . . . . . 8 | |
22 | 1st2nd2 6166 | . . . . . . . . . 10 | |
23 | 14, 22 | syl 14 | . . . . . . . . 9 |
24 | 23 | sneqd 3602 | . . . . . . . 8 |
25 | 21, 24 | eleqtrid 2264 | . . . . . . 7 |
26 | simprl 529 | . . . . . . . . 9 | |
27 | 23, 26 | eqeltrrd 2253 | . . . . . . . 8 |
28 | 27 | snssd 3734 | . . . . . . 7 |
29 | xpeq1 4634 | . . . . . . . . . 10 | |
30 | 29 | eleq2d 2245 | . . . . . . . . 9 |
31 | 29 | sseq1d 3182 | . . . . . . . . 9 |
32 | 30, 31 | anbi12d 473 | . . . . . . . 8 |
33 | xpeq2 4635 | . . . . . . . . . . 11 | |
34 | 1stexg 6158 | . . . . . . . . . . . . 13 | |
35 | 34 | elv 2739 | . . . . . . . . . . . 12 |
36 | 2ndexg 6159 | . . . . . . . . . . . . 13 | |
37 | 36 | elv 2739 | . . . . . . . . . . . 12 |
38 | 35, 37 | xpsn 5684 | . . . . . . . . . . 11 |
39 | 33, 38 | eqtrdi 2224 | . . . . . . . . . 10 |
40 | 39 | eleq2d 2245 | . . . . . . . . 9 |
41 | 39 | sseq1d 3182 | . . . . . . . . 9 |
42 | 40, 41 | anbi12d 473 | . . . . . . . 8 |
43 | 32, 42 | rspc2ev 2854 | . . . . . . 7 |
44 | 17, 20, 25, 28, 43 | syl112anc 1242 | . . . . . 6 |
45 | 44 | expr 375 | . . . . 5 |
46 | 45 | ralrimdva 2555 | . . . 4 |
47 | eltx 13339 | . . . . 5 | |
48 | 1, 2, 47 | syl2an 289 | . . . 4 |
49 | 46, 48 | sylibrd 169 | . . 3 |
50 | 49 | ssrdv 3159 | . 2 |
51 | 12, 50 | eqssd 3170 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 wceq 1353 wcel 2146 wral 2453 wrex 2454 cvv 2735 wss 3127 cpw 3572 csn 3589 cop 3592 cuni 3805 cxp 4618 cfv 5208 (class class class)co 5865 c1st 6129 c2nd 6130 ctop 13075 ctx 13332 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-topgen 12640 df-top 13076 df-topon 13089 df-bases 13121 df-tx 13333 |
This theorem is referenced by: (None) |
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