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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 8770 | . . . . . . 7 |
3 | 2 | simplbi 259 | . . . . . 6 |
4 | 3 | adantl 262 | . . . . 5 |
5 | 4 | simp3d 918 | . . . 4 |
6 | simplrl 487 | . . . . 5 | |
7 | 2 | simprbi 260 | . . . . . . 7 |
8 | 7 | adantl 262 | . . . . . 6 |
9 | 8 | simpld 105 | . . . . 5 |
10 | simplll 485 | . . . . . 6 | |
11 | 4 | simp1d 916 | . . . . . 6 |
12 | ixxss12.3 | . . . . . 6 | |
13 | 10, 11, 5, 12 | syl3anc 1135 | . . . . 5 |
14 | 6, 9, 13 | mp2and 409 | . . . 4 |
15 | 8 | simprd 107 | . . . . 5 |
16 | simplrr 488 | . . . . 5 | |
17 | 4 | simp2d 917 | . . . . . 6 |
18 | simpllr 486 | . . . . . 6 | |
19 | ixxss12.4 | . . . . . 6 | |
20 | 5, 17, 18, 19 | syl3anc 1135 | . . . . 5 |
21 | 15, 16, 20 | mp2and 409 | . . . 4 |
22 | ixxssixx.1 | . . . . . 6 | |
23 | 22 | elixx1 8766 | . . . . 5 |
24 | 23 | ad2antrr 457 | . . . 4 |
25 | 5, 14, 21, 24 | mpbir3and 1087 | . . 3 |
26 | 25 | ex 108 | . 2 |
27 | 26 | ssrdv 2951 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wceq 1243 wcel 1393 crab 2310 wss 2917 class class class wbr 3764 (class class class)co 5512 cmpt2 5514 cxr 7059 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-pnf 7062 df-mnf 7063 df-xr 7064 |
This theorem is referenced by: iccss 8810 iccssioo 8811 icossico 8812 iccss2 8813 iccssico 8814 iocssioo 8832 icossioo 8833 ioossioo 8834 |
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