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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 12254 | . . . . 5 | |
2 | distop 12254 | . . . . 5 | |
3 | unipw 4139 | . . . . . . 7 | |
4 | 3 | eqcomi 2143 | . . . . . 6 |
5 | unipw 4139 | . . . . . . 7 | |
6 | 5 | eqcomi 2143 | . . . . . 6 |
7 | 4, 6 | txuni 12432 | . . . . 5 |
8 | 1, 2, 7 | syl2an 287 | . . . 4 |
9 | eqimss2 3152 | . . . 4 | |
10 | 8, 9 | syl 14 | . . 3 |
11 | sspwuni 3897 | . . 3 | |
12 | 10, 11 | sylibr 133 | . 2 |
13 | elelpwi 3522 | . . . . . . . . 9 | |
14 | 13 | adantl 275 | . . . . . . . 8 |
15 | xp1st 6063 | . . . . . . . 8 | |
16 | snelpwi 4134 | . . . . . . . 8 | |
17 | 14, 15, 16 | 3syl 17 | . . . . . . 7 |
18 | xp2nd 6064 | . . . . . . . 8 | |
19 | snelpwi 4134 | . . . . . . . 8 | |
20 | 14, 18, 19 | 3syl 17 | . . . . . . 7 |
21 | vsnid 3557 | . . . . . . . 8 | |
22 | 1st2nd2 6073 | . . . . . . . . . 10 | |
23 | 14, 22 | syl 14 | . . . . . . . . 9 |
24 | 23 | sneqd 3540 | . . . . . . . 8 |
25 | 21, 24 | eleqtrid 2228 | . . . . . . 7 |
26 | simprl 520 | . . . . . . . . 9 | |
27 | 23, 26 | eqeltrrd 2217 | . . . . . . . 8 |
28 | 27 | snssd 3665 | . . . . . . 7 |
29 | xpeq1 4553 | . . . . . . . . . 10 | |
30 | 29 | eleq2d 2209 | . . . . . . . . 9 |
31 | 29 | sseq1d 3126 | . . . . . . . . 9 |
32 | 30, 31 | anbi12d 464 | . . . . . . . 8 |
33 | xpeq2 4554 | . . . . . . . . . . 11 | |
34 | 1stexg 6065 | . . . . . . . . . . . . 13 | |
35 | 34 | elv 2690 | . . . . . . . . . . . 12 |
36 | 2ndexg 6066 | . . . . . . . . . . . . 13 | |
37 | 36 | elv 2690 | . . . . . . . . . . . 12 |
38 | 35, 37 | xpsn 5596 | . . . . . . . . . . 11 |
39 | 33, 38 | syl6eq 2188 | . . . . . . . . . 10 |
40 | 39 | eleq2d 2209 | . . . . . . . . 9 |
41 | 39 | sseq1d 3126 | . . . . . . . . 9 |
42 | 40, 41 | anbi12d 464 | . . . . . . . 8 |
43 | 32, 42 | rspc2ev 2804 | . . . . . . 7 |
44 | 17, 20, 25, 28, 43 | syl112anc 1220 | . . . . . 6 |
45 | 44 | expr 372 | . . . . 5 |
46 | 45 | ralrimdva 2512 | . . . 4 |
47 | eltx 12428 | . . . . 5 | |
48 | 1, 2, 47 | syl2an 287 | . . . 4 |
49 | 46, 48 | sylibrd 168 | . . 3 |
50 | 49 | ssrdv 3103 | . 2 |
51 | 12, 50 | eqssd 3114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 wrex 2417 cvv 2686 wss 3071 cpw 3510 csn 3527 cop 3530 cuni 3736 cxp 4537 cfv 5123 (class class class)co 5774 c1st 6036 c2nd 6037 ctop 12164 ctx 12421 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-topgen 12141 df-top 12165 df-topon 12178 df-bases 12210 df-tx 12422 |
This theorem is referenced by: (None) |
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