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Mirrors > Home > ILE Home > Th. List > setsmsbasg | Unicode version |
Description: The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
setsms.x | |
setsms.d | |
setsms.k | sSet TopSet |
setsmsbasg.m | |
setsmsbasg.d |
Ref | Expression |
---|---|
setsmsbasg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsmsbasg.m | . . 3 | |
2 | setsmsbasg.d | . . 3 | |
3 | baseslid 12393 | . . . 4 Slot | |
4 | 1re 7889 | . . . . . 6 | |
5 | 1lt9 9052 | . . . . . 6 | |
6 | 4, 5 | ltneii 7986 | . . . . 5 |
7 | basendx 12391 | . . . . . 6 | |
8 | tsetndx 12485 | . . . . . 6 TopSet | |
9 | 7, 8 | neeq12i 2351 | . . . . 5 TopSet |
10 | 6, 9 | mpbir 145 | . . . 4 TopSet |
11 | 9nn 9016 | . . . . 5 | |
12 | 8, 11 | eqeltri 2237 | . . . 4 TopSet |
13 | 3, 10, 12 | setsslnid 12388 | . . 3 sSet TopSet |
14 | 1, 2, 13 | syl2anc 409 | . 2 sSet TopSet |
15 | setsms.x | . 2 | |
16 | setsms.k | . . 3 sSet TopSet | |
17 | 16 | fveq2d 5484 | . 2 sSet TopSet |
18 | 14, 15, 17 | 3eqtr4d 2207 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1342 wcel 2135 wne 2334 cop 3573 cxp 4596 cres 4600 cfv 5182 (class class class)co 5836 c1 7745 cn 8848 c9 8906 cnx 12334 sSet csts 12335 cbs 12337 TopSetcts 12405 cds 12408 cmopn 12532 |
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-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-pre-ltirr 7856 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
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-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 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-iota 5147 df-fun 5184 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-ltxr 7929 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-9 8914 df-ndx 12340 df-slot 12341 df-base 12343 df-sets 12344 df-tset 12418 |
This theorem is referenced by: (None) |
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