![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pltne | Structured version Visualization version GIF version |
Description: The "less than" relation is not reflexive. (df-pss 3934 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 2737 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
2 | pltne.s | . . . 4 β’ < = (ltβπΎ) | |
3 | 1, 2 | pltval 18228 | . . 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 501 | . 2 β’ (((πΎ β π΄ β§ π β π΅ β§ π β πΆ) β§ π < π) β π β π) |
5 | 4 | ex 414 | 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 1088 = wceq 1542 β wcel 2107 β wne 2944 class class class wbr 5110 βcfv 6501 lecple 17147 ltcplt 18204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-plt 18226 |
This theorem is referenced by: pltirr 18231 ogrpaddlt 31967 ornglmullt 32142 orngrmullt 32143 ofldchr 32149 isarchiofld 32152 atlen0 37801 1cvratex 37965 ps-2 37970 lhpn0 38496 |
Copyright terms: Public domain | W3C validator |