Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > blssexps | Unicode version |
Description: Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
Ref | Expression |
---|---|
blssexps | PsMet |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | blssps 13186 | . . . . . . 7 PsMet | |
2 | sstr 3155 | . . . . . . . . 9 | |
3 | 2 | expcom 115 | . . . . . . . 8 |
4 | 3 | reximdv 2571 | . . . . . . 7 |
5 | 1, 4 | syl5com 29 | . . . . . 6 PsMet |
6 | 5 | 3expa 1198 | . . . . 5 PsMet |
7 | 6 | expimpd 361 | . . . 4 PsMet |
8 | 7 | adantlr 474 | . . 3 PsMet |
9 | 8 | rexlimdva 2587 | . 2 PsMet |
10 | simpll 524 | . . . . 5 PsMet PsMet | |
11 | simplr 525 | . . . . 5 PsMet | |
12 | rpxr 9611 | . . . . . 6 | |
13 | 12 | ad2antrl 487 | . . . . 5 PsMet |
14 | blelrnps 13178 | . . . . 5 PsMet | |
15 | 10, 11, 13, 14 | syl3anc 1233 | . . . 4 PsMet |
16 | simprl 526 | . . . . 5 PsMet | |
17 | blcntrps 13174 | . . . . 5 PsMet | |
18 | 10, 11, 16, 17 | syl3anc 1233 | . . . 4 PsMet |
19 | simprr 527 | . . . 4 PsMet | |
20 | eleq2 2234 | . . . . . 6 | |
21 | sseq1 3170 | . . . . . 6 | |
22 | 20, 21 | anbi12d 470 | . . . . 5 |
23 | 22 | rspcev 2834 | . . . 4 |
24 | 15, 18, 19, 23 | syl12anc 1231 | . . 3 PsMet |
25 | 24 | rexlimdvaa 2588 | . 2 PsMet |
26 | 9, 25 | impbid 128 | 1 PsMet |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wceq 1348 wcel 2141 wrex 2449 wss 3121 crn 4610 cfv 5196 (class class class)co 5851 cxr 7946 crp 9603 PsMetcpsmet 12738 cbl 12741 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 ax-pre-mulext 7885 ax-arch 7886 |
This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-po 4279 df-iso 4280 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-map 6626 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-reap 8487 df-ap 8494 df-div 8583 df-inn 8872 df-2 8930 df-n0 9129 df-z 9206 df-uz 9481 df-q 9572 df-rp 9604 df-xneg 9722 df-xadd 9723 df-psmet 12746 df-bl 12749 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |