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Mirrors > Home > ILE Home > Th. List > ixxdisj | Unicode version |
Description: Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.) |
Ref | Expression |
---|---|
ixxssixx.1 |
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ixxun.2 |
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ixxun.3 |
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Ref | Expression |
---|---|
ixxdisj |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3183 |
. . . 4
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2 | ixxssixx.1 |
. . . . . . . . . . 11
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3 | 2 | elixx1 9305 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | 3 | 3adant3 963 |
. . . . . . . . 9
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5 | 4 | biimpa 290 |
. . . . . . . 8
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6 | 5 | simp3d 957 |
. . . . . . 7
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7 | 6 | adantrr 463 |
. . . . . 6
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8 | ixxun.2 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 8 | elixx1 9305 |
. . . . . . . . . . 11
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10 | 9 | 3adant1 961 |
. . . . . . . . . 10
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11 | 10 | biimpa 290 |
. . . . . . . . 9
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12 | 11 | simp2d 956 |
. . . . . . . 8
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13 | simpl2 947 |
. . . . . . . . 9
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14 | 11 | simp1d 955 |
. . . . . . . . 9
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15 | ixxun.3 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 13, 14, 15 | syl2anc 403 |
. . . . . . . 8
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17 | 12, 16 | mpbid 145 |
. . . . . . 7
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18 | 17 | adantrl 462 |
. . . . . 6
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19 | 7, 18 | pm2.65da 622 |
. . . . 5
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20 | 19 | pm2.21d 584 |
. . . 4
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21 | 1, 20 | syl5bi 150 |
. . 3
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22 | 21 | ssrdv 3031 |
. 2
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23 | ss0 3322 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 22, 23 | syl 14 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3955 ax-pow 4007 ax-pr 4034 ax-un 4258 ax-setind 4351 ax-cnex 7426 ax-resscn 7427 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3429 df-sn 3450 df-pr 3451 df-op 3453 df-uni 3652 df-br 3844 df-opab 3898 df-id 4118 df-xp 4442 df-rel 4443 df-cnv 4444 df-co 4445 df-dm 4446 df-iota 4975 df-fun 5012 df-fv 5018 df-ov 5647 df-oprab 5648 df-mpt2 5649 df-pnf 7514 df-mnf 7515 df-xr 7516 |
This theorem is referenced by: ioodisj 9400 |
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