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Mirrors > Home > ILE Home > Th. List > ixxss2 | Unicode version |
Description: Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
ixxssixx.1 | |
ixxss2.2 | |
ixxss2.3 |
Ref | Expression |
---|---|
ixxss2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixxss2.2 | . . . . . . . 8 | |
2 | 1 | elixx3g 8924 | . . . . . . 7 |
3 | 2 | simplbi 268 | . . . . . 6 |
4 | 3 | adantl 271 | . . . . 5 |
5 | 4 | simp3d 952 | . . . 4 |
6 | 2 | simprbi 269 | . . . . . 6 |
7 | 6 | adantl 271 | . . . . 5 |
8 | 7 | simpld 110 | . . . 4 |
9 | 7 | simprd 112 | . . . . 5 |
10 | simplr 496 | . . . . 5 | |
11 | 4 | simp2d 951 | . . . . . 6 |
12 | simpll 495 | . . . . . 6 | |
13 | ixxss2.3 | . . . . . 6 | |
14 | 5, 11, 12, 13 | syl3anc 1169 | . . . . 5 |
15 | 9, 10, 14 | mp2and 423 | . . . 4 |
16 | 4 | simp1d 950 | . . . . 5 |
17 | ixxssixx.1 | . . . . . 6 | |
18 | 17 | elixx1 8920 | . . . . 5 |
19 | 16, 12, 18 | syl2anc 403 | . . . 4 |
20 | 5, 8, 15, 19 | mpbir3and 1121 | . . 3 |
21 | 20 | ex 113 | . 2 |
22 | 21 | ssrdv 3005 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wceq 1284 wcel 1433 crab 2352 wss 2973 class class class wbr 3785 (class class class)co 5532 cmpt2 5534 cxr 7152 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 |
This theorem is referenced by: iooss2 8940 |
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