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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 |
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ixxss2.2 |
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ixxss2.3 |
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Ref | Expression |
---|---|
ixxss2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixxss2.2 |
. . . . . . . 8
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2 | 1 | elixx3g 9899 |
. . . . . . 7
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3 | 2 | simplbi 274 |
. . . . . 6
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4 | 3 | adantl 277 |
. . . . 5
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5 | 4 | simp3d 1011 |
. . . 4
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6 | 2 | simprbi 275 |
. . . . . 6
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7 | 6 | adantl 277 |
. . . . 5
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8 | 7 | simpld 112 |
. . . 4
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9 | 7 | simprd 114 |
. . . . 5
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10 | simplr 528 |
. . . . 5
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11 | 4 | simp2d 1010 |
. . . . . 6
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12 | simpll 527 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | ixxss2.3 |
. . . . . 6
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14 | 5, 11, 12, 13 | syl3anc 1238 |
. . . . 5
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15 | 9, 10, 14 | mp2and 433 |
. . . 4
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16 | 4 | simp1d 1009 |
. . . . 5
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17 | ixxssixx.1 |
. . . . . 6
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18 | 17 | elixx1 9895 |
. . . . 5
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19 | 16, 12, 18 | syl2anc 411 |
. . . 4
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20 | 5, 8, 15, 19 | mpbir3and 1180 |
. . 3
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21 | 20 | ex 115 |
. 2
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22 | 21 | ssrdv 3161 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-xr 7994 |
This theorem is referenced by: iooss2 9915 |
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