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Mirrors > Home > MPE Home > Th. List > eltx | Structured version Visualization version Unicode version |
Description: A set in a product is open iff each point is surrounded by an open rectangle. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Ref | Expression |
---|---|
eltx |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . 4 | |
2 | 1 | txval 21367 | . . 3 |
3 | 2 | eleq2d 2687 | . 2 |
4 | 1 | txbasex 21369 | . . . 4 |
5 | eltg2b 20763 | . . . 4 | |
6 | 4, 5 | syl 17 | . . 3 |
7 | vex 3203 | . . . . . . 7 | |
8 | vex 3203 | . . . . . . 7 | |
9 | 7, 8 | xpex 6962 | . . . . . 6 |
10 | 9 | rgen2w 2925 | . . . . 5 |
11 | eqid 2622 | . . . . . 6 | |
12 | eleq2 2690 | . . . . . . 7 | |
13 | sseq1 3626 | . . . . . . 7 | |
14 | 12, 13 | anbi12d 747 | . . . . . 6 |
15 | 11, 14 | rexrnmpt2 6776 | . . . . 5 |
16 | 10, 15 | ax-mp 5 | . . . 4 |
17 | 16 | ralbii 2980 | . . 3 |
18 | 6, 17 | syl6bb 276 | . 2 |
19 | 3, 18 | bitrd 268 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 wrex 2913 cvv 3200 wss 3574 cxp 5112 crn 5115 cfv 5888 (class class class)co 6650 cmpt2 6652 ctg 16098 ctx 21363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-topgen 16104 df-tx 21365 |
This theorem is referenced by: txcls 21407 txcnpi 21411 txdis 21435 txindis 21437 txdis1cn 21438 txlly 21439 txnlly 21440 txtube 21443 txcmplem1 21444 hausdiag 21448 tx1stc 21453 qustgplem 21924 txomap 29901 cvmlift2lem10 31294 |
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