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 8552 | . . . . . 6 # | |
2 | 1 | 3expia 1200 | . . . . 5 # |
3 | recn 7907 | . . . . . 6 | |
4 | recn 7907 | . . . . . 6 | |
5 | apsym 8525 | . . . . . 6 # # | |
6 | 3, 4, 5 | syl2an 287 | . . . . 5 # # |
7 | 2, 6 | sylibrd 168 | . . . 4 # |
8 | orc 707 | . . . . 5 # # # | |
9 | df-dc 830 | . . . . 5 DECID # # # | |
10 | 8, 9 | sylibr 133 | . . . 4 # DECID # |
11 | 7, 10 | syl6 33 | . . 3 DECID # |
12 | apti 8541 | . . . . 5 # | |
13 | 3, 4, 12 | syl2an 287 | . . . 4 # |
14 | olc 706 | . . . . 5 # # # | |
15 | 14, 9 | sylibr 133 | . . . 4 # DECID # |
16 | 13, 15 | syl6bi 162 | . . 3 DECID # |
17 | ltap 8552 | . . . . . 6 # | |
18 | 17, 10 | syl 14 | . . . . 5 DECID # |
19 | 18 | 3expia 1200 | . . . 4 DECID # |
20 | 19 | ancoms 266 | . . 3 DECID # |
21 | 11, 16, 20 | 3jaod 1299 | . 2 DECID # |
22 | reaplt 8507 | . . . . 5 # | |
23 | orc 707 | . . . . . . 7 | |
24 | 23 | orim1i 755 | . . . . . 6 |
25 | df-3or 974 | . . . . . 6 | |
26 | 24, 25 | sylibr 133 | . . . . 5 |
27 | 22, 26 | syl6bi 162 | . . . 4 # |
28 | 3mix2 1162 | . . . . 5 | |
29 | 13, 28 | syl6bir 163 | . . . 4 # |
30 | 27, 29 | jaod 712 | . . 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 703 DECID wdc 829 w3o 972 w3a 973 wceq 1348 wcel 2141 class class class wbr 3989 cc 7772 cr 7773 clt 7954 # cap 8500 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-ltxr 7959 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 |
This theorem is referenced by: reap0 14090 |
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