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| Mirrors > Home > MPE Home > Th. List > pltnle | Structured version Visualization version GIF version | ||
| Description: "Less than" implies not converse "less than or equal to". (Contributed by NM, 18-Oct-2011.) |
| Ref | Expression |
|---|---|
| pleval2.b | β’ π΅ = (BaseβπΎ) |
| pleval2.l | β’ β€ = (leβπΎ) |
| pleval2.s | β’ < = (ltβπΎ) |
| Ref | Expression |
|---|---|
| pltnle | β’ (((πΎ β Poset β§ π β π΅ β§ π β π΅) β§ π < π) β Β¬ π β€ π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pleval2.l | . . . 4 β’ β€ = (leβπΎ) | |
| 2 | pleval2.s | . . . 4 β’ < = (ltβπΎ) | |
| 3 | 1, 2 | pltval 18289 | . . 3 β’ ((πΎ β Poset β§ π β π΅ β§ π β π΅) β (π < π β (π β€ π β§ π β π))) |
| 4 | pleval2.b | . . . . . . . 8 β’ π΅ = (BaseβπΎ) | |
| 5 | 4, 1 | posasymb 18276 | . . . . . . 7 β’ ((πΎ β Poset β§ π β π΅ β§ π β π΅) β ((π β€ π β§ π β€ π) β π = π)) |
| 6 | 5 | biimpd 228 | . . . . . 6 β’ ((πΎ β Poset β§ π β π΅ β§ π β π΅) β ((π β€ π β§ π β€ π) β π = π)) |
| 7 | 6 | expdimp 453 | . . . . 5 β’ (((πΎ β Poset β§ π β π΅ β§ π β π΅) β§ π β€ π) β (π β€ π β π = π)) |
| 8 | 7 | necon3ad 2953 | . . . 4 β’ (((πΎ β Poset β§ π β π΅ β§ π β π΅) β§ π β€ π) β (π β π β Β¬ π β€ π)) |
| 9 | 8 | expimpd 454 | . . 3 β’ ((πΎ β Poset β§ π β π΅ β§ π β π΅) β ((π β€ π β§ π β π) β Β¬ π β€ π)) |
| 10 | 3, 9 | sylbid 239 | . 2 β’ ((πΎ β Poset β§ π β π΅ β§ π β π΅) β (π < π β Β¬ π β€ π)) |
| 11 | 10 | imp 407 | 1 β’ (((πΎ β Poset β§ π β π΅ β§ π β π΅) β§ π < π) β Β¬ π β€ π) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2940 class class class wbr 5148 βcfv 6543 Basecbs 17148 lecple 17208 Posetcpo 18264 ltcplt 18265 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-proset 18252 df-poset 18270 df-plt 18287 |
| This theorem is referenced by: pltnlt 18297 pltn2lp 18298 ncvr1 38445 cvrnle 38453 atlrelat1 38494 cvrat 38596 |
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