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Mirrors > Home > MPE Home > Th. List > pltne | Structured version Visualization version GIF version |
Description: The "less than" relation is not reflexive. (df-pss 3966 analog.) (Contributed by NM, 2-Dec-2011.) |
Ref | Expression |
---|---|
pltne.s | β’ < = (ltβπΎ) |
Ref | Expression |
---|---|
pltne | β’ ((πΎ β π΄ β§ π β π΅ β§ π β πΆ) β (π < π β π β π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
2 | pltne.s | . . . 4 β’ < = (ltβπΎ) | |
3 | 1, 2 | pltval 18323 | . . 3 β’ ((πΎ β π΄ β§ π β π΅ β§ π β πΆ) β (π < π β (π(leβπΎ)π β§ π β π))) |
4 | 3 | simplbda 499 | . 2 β’ (((πΎ β π΄ β§ π β π΅ β§ π β πΆ) β§ π < π) β π β π) |
5 | 4 | ex 412 | 1 β’ ((πΎ β π΄ β§ π β π΅ β§ π β πΆ) β (π < π β π β π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1085 = wceq 1534 β wcel 2099 β wne 2937 class class class wbr 5148 βcfv 6548 lecple 17239 ltcplt 18299 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-plt 18321 |
This theorem is referenced by: pltirr 18326 ogrpaddlt 32797 ornglmullt 33022 orngrmullt 33023 ofldchr 33029 isarchiofld 33032 atlen0 38782 1cvratex 38946 ps-2 38951 lhpn0 39477 |
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