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Mirrors > Home > MPE Home > Th. List > axpre-lttri | Structured version Visualization version Unicode version |
Description: Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttri 10109. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 10010. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axpre-lttri |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elreal 9952 | . 2 | |
2 | elreal 9952 | . 2 | |
3 | breq1 4656 | . . 3 | |
4 | eqeq1 2626 | . . . . 5 | |
5 | breq2 4657 | . . . . 5 | |
6 | 4, 5 | orbi12d 746 | . . . 4 |
7 | 6 | notbid 308 | . . 3 |
8 | 3, 7 | bibi12d 335 | . 2 |
9 | breq2 4657 | . . 3 | |
10 | eqeq2 2633 | . . . . 5 | |
11 | breq1 4656 | . . . . 5 | |
12 | 10, 11 | orbi12d 746 | . . . 4 |
13 | 12 | notbid 308 | . . 3 |
14 | 9, 13 | bibi12d 335 | . 2 |
15 | ltsosr 9915 | . . . 4 | |
16 | sotric 5061 | . . . 4 | |
17 | 15, 16 | mpan 706 | . . 3 |
18 | ltresr 9961 | . . 3 | |
19 | vex 3203 | . . . . . 6 | |
20 | 19 | eqresr 9958 | . . . . 5 |
21 | ltresr 9961 | . . . . 5 | |
22 | 20, 21 | orbi12i 543 | . . . 4 |
23 | 22 | notbii 310 | . . 3 |
24 | 17, 18, 23 | 3bitr4g 303 | . 2 |
25 | 1, 2, 8, 14, 24 | 2gencl 3236 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 wceq 1483 wcel 1990 cop 4183 class class class wbr 4653 wor 5034 cnr 9687 c0r 9688 cltr 9693 cr 9935 cltrr 9940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-ec 7744 df-qs 7748 df-ni 9694 df-pli 9695 df-mi 9696 df-lti 9697 df-plpq 9730 df-mpq 9731 df-ltpq 9732 df-enq 9733 df-nq 9734 df-erq 9735 df-plq 9736 df-mq 9737 df-1nq 9738 df-rq 9739 df-ltnq 9740 df-np 9803 df-1p 9804 df-plp 9805 df-ltp 9807 df-enr 9877 df-nr 9878 df-ltr 9881 df-0r 9882 df-r 9946 df-lt 9949 |
This theorem is referenced by: (None) |
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