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Mirrors > Home > MPE Home > Th. List > pltval | Structured version Visualization version GIF version |
Description: Less-than relation. (df-pss 3930 analog.) (Contributed by NM, 12-Oct-2011.) |
Ref | Expression |
---|---|
pltval.l | β’ β€ = (leβπΎ) |
pltval.s | β’ < = (ltβπΎ) |
Ref | Expression |
---|---|
pltval | β’ ((πΎ β π΄ β§ π β π΅ β§ π β πΆ) β (π < π β (π β€ π β§ π β π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pltval.l | . . . . 5 β’ β€ = (leβπΎ) | |
2 | pltval.s | . . . . 5 β’ < = (ltβπΎ) | |
3 | 1, 2 | pltfval 18225 | . . . 4 β’ (πΎ β π΄ β < = ( β€ β I )) |
4 | 3 | breqd 5117 | . . 3 β’ (πΎ β π΄ β (π < π β π( β€ β I )π)) |
5 | brdif 5159 | . . . 4 β’ (π( β€ β I )π β (π β€ π β§ Β¬ π I π)) | |
6 | ideqg 5808 | . . . . . . 7 β’ (π β πΆ β (π I π β π = π)) | |
7 | 6 | necon3bbid 2978 | . . . . . 6 β’ (π β πΆ β (Β¬ π I π β π β π)) |
8 | 7 | adantl 483 | . . . . 5 β’ ((π β π΅ β§ π β πΆ) β (Β¬ π I π β π β π)) |
9 | 8 | anbi2d 630 | . . . 4 β’ ((π β π΅ β§ π β πΆ) β ((π β€ π β§ Β¬ π I π) β (π β€ π β§ π β π))) |
10 | 5, 9 | bitrid 283 | . . 3 β’ ((π β π΅ β§ π β πΆ) β (π( β€ β I )π β (π β€ π β§ π β π))) |
11 | 4, 10 | sylan9bb 511 | . 2 β’ ((πΎ β π΄ β§ (π β π΅ β§ π β πΆ)) β (π < π β (π β€ π β§ π β π))) |
12 | 11 | 3impb 1116 | 1 β’ ((πΎ β π΄ β§ π β π΅ β§ π β πΆ) β (π < π β (π β€ π β§ π β π))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2940 β cdif 3908 class class class wbr 5106 I cid 5531 βcfv 6497 lecple 17145 ltcplt 18202 |
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 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-plt 18224 |
This theorem is referenced by: pltle 18227 pltne 18228 pleval2i 18230 pltnle 18232 pltval3 18233 plttr 18236 latnlemlt 18366 latnle 18367 ipolt 18429 ogrpaddlt 31974 ogrpsublt 31978 ornglmullt 32149 orngrmullt 32150 orngmullt 32151 ofldlt1 32155 opltn0 37698 cvrval2 37782 cvrnbtwn2 37783 cvrnbtwn3 37784 cvrle 37786 cvrnbtwn4 37787 cvrne 37789 atlltn0 37814 hlrelat5N 37910 llnle 38027 lplnle 38049 llncvrlpln2 38066 lplncvrlvol2 38124 lhp2lt 38510 lautlt 38600 |
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