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Mirrors > Home > ILE Home > Th. List > ixxss12 | Unicode version |
Description: Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 20-Feb-2015.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
ixxssixx.1 | |
ixxss12.2 | |
ixxss12.3 | |
ixxss12.4 |
Ref | Expression |
---|---|
ixxss12 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixxss12.2 | . . . . . . . 8 | |
2 | 1 | elixx3g 9845 | . . . . . . 7 |
3 | 2 | simplbi 272 | . . . . . 6 |
4 | 3 | adantl 275 | . . . . 5 |
5 | 4 | simp3d 1006 | . . . 4 |
6 | simplrl 530 | . . . . 5 | |
7 | 2 | simprbi 273 | . . . . . . 7 |
8 | 7 | adantl 275 | . . . . . 6 |
9 | 8 | simpld 111 | . . . . 5 |
10 | simplll 528 | . . . . . 6 | |
11 | 4 | simp1d 1004 | . . . . . 6 |
12 | ixxss12.3 | . . . . . 6 | |
13 | 10, 11, 5, 12 | syl3anc 1233 | . . . . 5 |
14 | 6, 9, 13 | mp2and 431 | . . . 4 |
15 | 8 | simprd 113 | . . . . 5 |
16 | simplrr 531 | . . . . 5 | |
17 | 4 | simp2d 1005 | . . . . . 6 |
18 | simpllr 529 | . . . . . 6 | |
19 | ixxss12.4 | . . . . . 6 | |
20 | 5, 17, 18, 19 | syl3anc 1233 | . . . . 5 |
21 | 15, 16, 20 | mp2and 431 | . . . 4 |
22 | ixxssixx.1 | . . . . . 6 | |
23 | 22 | elixx1 9841 | . . . . 5 |
24 | 23 | ad2antrr 485 | . . . 4 |
25 | 5, 14, 21, 24 | mpbir3and 1175 | . . 3 |
26 | 25 | ex 114 | . 2 |
27 | 26 | ssrdv 3153 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wceq 1348 wcel 2141 crab 2452 wss 3121 class class class wbr 3987 (class class class)co 5850 cmpo 5852 cxr 7940 |
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 7852 ax-resscn 7853 |
This theorem depends on definitions: df-bi 116 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-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-ov 5853 df-oprab 5854 df-mpo 5855 df-pnf 7943 df-mnf 7944 df-xr 7945 |
This theorem is referenced by: iccss 9885 iccssioo 9886 icossico 9887 iccss2 9888 iccssico 9889 iocssioo 9907 icossioo 9908 ioossioo 9909 |
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