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Mirrors > Home > MPE Home > Th. List > ixxss1 | Structured version Visualization version 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 |
---|---|
ixx.1 | |
ixxss1.2 | |
ixxss1.3 |
Ref | Expression |
---|---|
ixxss1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixxss1.2 | . . . . . . . 8 | |
2 | 1 | elixx3g 12188 | . . . . . . 7 |
3 | 2 | simplbi 476 | . . . . . 6 |
4 | 3 | adantl 482 | . . . . 5 |
5 | 4 | simp3d 1075 | . . . 4 |
6 | simplr 792 | . . . . 5 | |
7 | 2 | simprbi 480 | . . . . . . 7 |
8 | 7 | adantl 482 | . . . . . 6 |
9 | 8 | simpld 475 | . . . . 5 |
10 | simpll 790 | . . . . . 6 | |
11 | 4 | simp1d 1073 | . . . . . 6 |
12 | ixxss1.3 | . . . . . 6 | |
13 | 10, 11, 5, 12 | syl3anc 1326 | . . . . 5 |
14 | 6, 9, 13 | mp2and 715 | . . . 4 |
15 | 8 | simprd 479 | . . . 4 |
16 | 4 | simp2d 1074 | . . . . 5 |
17 | ixx.1 | . . . . . 6 | |
18 | 17 | elixx1 12184 | . . . . 5 |
19 | 10, 16, 18 | syl2anc 693 | . . . 4 |
20 | 5, 14, 15, 19 | mpbir3and 1245 | . . 3 |
21 | 20 | ex 450 | . 2 |
22 | 21 | ssrdv 3609 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 crab 2916 wss 3574 class class class wbr 4653 (class class class)co 6650 cmpt2 6652 cxr 10073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-xr 10078 |
This theorem is referenced by: iooss1 12210 limsupgord 14203 pnfnei 21024 dvfsumrlimge0 23793 dvfsumrlim2 23795 tanord1 24283 rlimcnp 24692 rlimcnp2 24693 dchrisum0lem2a 25206 pntleml 25300 pnt 25303 liminfgord 39986 |
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