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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 3960 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 2724 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
2 | pltne.s | . . . 4 β’ < = (ltβπΎ) | |
3 | 1, 2 | pltval 18293 | . . 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 1084 = wceq 1533 β wcel 2098 β wne 2932 class class class wbr 5139 βcfv 6534 lecple 17209 ltcplt 18269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-iota 6486 df-fun 6536 df-fv 6542 df-plt 18291 |
This theorem is referenced by: pltirr 18296 ogrpaddlt 32728 ornglmullt 32917 orngrmullt 32918 ofldchr 32924 isarchiofld 32927 atlen0 38683 1cvratex 38847 ps-2 38852 lhpn0 39378 |
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