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Mirrors > Home > MPE Home > Th. List > pltval | Structured version Visualization version GIF version |
Description: Less-than relation. (df-pss 3963 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 18314 | . . . 4 β’ (πΎ β π΄ β < = ( β€ β I )) |
4 | 3 | breqd 5153 | . . 3 β’ (πΎ β π΄ β (π < π β π( β€ β I )π)) |
5 | brdif 5195 | . . . 4 β’ (π( β€ β I )π β (π β€ π β§ Β¬ π I π)) | |
6 | ideqg 5848 | . . . . . . 7 β’ (π β πΆ β (π I π β π = π)) | |
7 | 6 | necon3bbid 2973 | . . . . . 6 β’ (π β πΆ β (Β¬ π I π β π β π)) |
8 | 7 | adantl 481 | . . . . 5 β’ ((π β π΅ β§ π β πΆ) β (Β¬ π I π β π β π)) |
9 | 8 | anbi2d 628 | . . . 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 509 | . 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 1113 | 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 395 β§ w3a 1085 = wceq 1534 β wcel 2099 β wne 2935 β cdif 3941 class class class wbr 5142 I cid 5569 βcfv 6542 lecple 17231 ltcplt 18291 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-plt 18313 |
This theorem is referenced by: pltle 18316 pltne 18317 pleval2i 18319 pltnle 18321 pltval3 18322 plttr 18325 latnlemlt 18455 latnle 18456 ipolt 18518 ogrpaddlt 32775 ogrpsublt 32779 ornglmullt 32962 orngrmullt 32963 orngmullt 32964 ofldlt1 32968 opltn0 38599 cvrval2 38683 cvrnbtwn2 38684 cvrnbtwn3 38685 cvrle 38687 cvrnbtwn4 38688 cvrne 38690 atlltn0 38715 hlrelat5N 38811 llnle 38928 lplnle 38950 llncvrlpln2 38967 lplncvrlvol2 39025 lhp2lt 39411 lautlt 39501 |
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