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Mirrors > Home > MPE Home > Th. List > lttri4 | Structured version Unicode version |
Description: Trichotomy law for 'less than'. (Contributed by NM, 20-Sep-2007.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Ref | Expression |
---|---|
lttri4 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltso 9567 |
. 2
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2 | solin 4773 |
. 2
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3 | 1, 2 | mpan 670 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4522 ax-nul 4530 ax-pow 4579 ax-pr 4640 ax-un 6483 ax-resscn 9451 ax-pre-lttri 9468 ax-pre-lttrn 9469 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3397 df-dif 3440 df-un 3442 df-in 3444 df-ss 3451 df-nul 3747 df-if 3901 df-pw 3971 df-sn 3987 df-pr 3989 df-op 3993 df-uni 4201 df-br 4402 df-opab 4460 df-mpt 4461 df-id 4745 df-po 4750 df-so 4751 df-xp 4955 df-rel 4956 df-cnv 4957 df-co 4958 df-dm 4959 df-rn 4960 df-res 4961 df-ima 4962 df-iota 5490 df-fun 5529 df-fn 5530 df-f 5531 df-f1 5532 df-fo 5533 df-f1o 5534 df-fv 5535 df-er 7212 df-en 7422 df-dom 7423 df-sdom 7424 df-pnf 9532 df-mnf 9533 df-ltxr 9535 |
This theorem is referenced by: lttri4d 9627 xlemul1a 11363 xadddi 11370 mbfmulc2lem 21259 c1lip1 21603 reeff1o 22046 tanabsge 22102 logcnlem3 22223 atantan 22452 atanbnd 22455 |
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