Mathbox for Jim Kingdon |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > Mathboxes > triap | Unicode version |
Description: Two ways of stating real number trichotomy. (Contributed by Jim Kingdon, 23-Aug-2023.) |
Ref | Expression |
---|---|
triap | DECID # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltap 8395 | . . . . . 6 # | |
2 | 1 | 3expia 1183 | . . . . 5 # |
3 | recn 7753 | . . . . . 6 | |
4 | recn 7753 | . . . . . 6 | |
5 | apsym 8368 | . . . . . 6 # # | |
6 | 3, 4, 5 | syl2an 287 | . . . . 5 # # |
7 | 2, 6 | sylibrd 168 | . . . 4 # |
8 | orc 701 | . . . . 5 # # # | |
9 | df-dc 820 | . . . . 5 DECID # # # | |
10 | 8, 9 | sylibr 133 | . . . 4 # DECID # |
11 | 7, 10 | syl6 33 | . . 3 DECID # |
12 | apti 8384 | . . . . 5 # | |
13 | 3, 4, 12 | syl2an 287 | . . . 4 # |
14 | olc 700 | . . . . 5 # # # | |
15 | 14, 9 | sylibr 133 | . . . 4 # DECID # |
16 | 13, 15 | syl6bi 162 | . . 3 DECID # |
17 | ltap 8395 | . . . . . 6 # | |
18 | 17, 10 | syl 14 | . . . . 5 DECID # |
19 | 18 | 3expia 1183 | . . . 4 DECID # |
20 | 19 | ancoms 266 | . . 3 DECID # |
21 | 11, 16, 20 | 3jaod 1282 | . 2 DECID # |
22 | reaplt 8350 | . . . . 5 # | |
23 | orc 701 | . . . . . . 7 | |
24 | 23 | orim1i 749 | . . . . . 6 |
25 | df-3or 963 | . . . . . 6 | |
26 | 24, 25 | sylibr 133 | . . . . 5 |
27 | 22, 26 | syl6bi 162 | . . . 4 # |
28 | 3mix2 1151 | . . . . 5 | |
29 | 13, 28 | syl6bir 163 | . . . 4 # |
30 | 27, 29 | jaod 706 | . . 3 # # |
31 | 9, 30 | syl5bi 151 | . 2 DECID # |
32 | 21, 31 | impbid 128 | 1 DECID # |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 DECID wdc 819 w3o 961 w3a 962 wceq 1331 wcel 1480 class class class wbr 3929 cc 7618 cr 7619 clt 7800 # cap 8343 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |