Mathbox for Jim Kingdon |
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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 8508 | . . . . . 6 # | |
2 | 1 | 3expia 1187 | . . . . 5 # |
3 | recn 7865 | . . . . . 6 | |
4 | recn 7865 | . . . . . 6 | |
5 | apsym 8481 | . . . . . 6 # # | |
6 | 3, 4, 5 | syl2an 287 | . . . . 5 # # |
7 | 2, 6 | sylibrd 168 | . . . 4 # |
8 | orc 702 | . . . . 5 # # # | |
9 | df-dc 821 | . . . . 5 DECID # # # | |
10 | 8, 9 | sylibr 133 | . . . 4 # DECID # |
11 | 7, 10 | syl6 33 | . . 3 DECID # |
12 | apti 8497 | . . . . 5 # | |
13 | 3, 4, 12 | syl2an 287 | . . . 4 # |
14 | olc 701 | . . . . 5 # # # | |
15 | 14, 9 | sylibr 133 | . . . 4 # DECID # |
16 | 13, 15 | syl6bi 162 | . . 3 DECID # |
17 | ltap 8508 | . . . . . 6 # | |
18 | 17, 10 | syl 14 | . . . . 5 DECID # |
19 | 18 | 3expia 1187 | . . . 4 DECID # |
20 | 19 | ancoms 266 | . . 3 DECID # |
21 | 11, 16, 20 | 3jaod 1286 | . 2 DECID # |
22 | reaplt 8463 | . . . . 5 # | |
23 | orc 702 | . . . . . . 7 | |
24 | 23 | orim1i 750 | . . . . . 6 |
25 | df-3or 964 | . . . . . 6 | |
26 | 24, 25 | sylibr 133 | . . . . 5 |
27 | 22, 26 | syl6bi 162 | . . . 4 # |
28 | 3mix2 1152 | . . . . 5 | |
29 | 13, 28 | syl6bir 163 | . . . 4 # |
30 | 27, 29 | jaod 707 | . . 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 698 DECID wdc 820 w3o 962 w3a 963 wceq 1335 wcel 2128 class class class wbr 3965 cc 7730 cr 7731 clt 7912 # cap 8456 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-pnf 7914 df-mnf 7915 df-ltxr 7917 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 |
This theorem is referenced by: reap0 13629 |
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