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Mirrors > Home > ILE Home > Th. List > reapti | Unicode version |
Description: Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8408. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
reapti |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnr 7865 |
. . . . 5
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2 | 1 | adantr 274 |
. . . 4
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3 | oridm 747 |
. . . . . 6
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4 | breq2 3941 |
. . . . . . 7
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5 | breq1 3940 |
. . . . . . 7
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6 | 4, 5 | orbi12d 783 |
. . . . . 6
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7 | 3, 6 | bitr3id 193 |
. . . . 5
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8 | 7 | notbid 657 |
. . . 4
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9 | 2, 8 | syl5ibcom 154 |
. . 3
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10 | reapval 8362 |
. . . 4
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11 | 10 | notbid 657 |
. . 3
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12 | 9, 11 | sylibrd 168 |
. 2
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13 | axapti 7859 |
. . . 4
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14 | 13 | 3expia 1184 |
. . 3
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15 | 11, 14 | sylbid 149 |
. 2
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16 | 12, 15 | impbid 128 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-pre-ltirr 7756 ax-pre-apti 7759 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-pnf 7826 df-mnf 7827 df-ltxr 7829 df-reap 8361 |
This theorem is referenced by: rimul 8371 apreap 8373 apti 8408 |
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