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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvrne | Structured version Visualization version GIF version |
Description: The covers relation implies inequality. (Contributed by NM, 13-Oct-2011.) |
Ref | Expression |
---|---|
cvrne.b | β’ π΅ = (BaseβπΎ) |
cvrne.c | β’ πΆ = ( β βπΎ) |
Ref | Expression |
---|---|
cvrne | β’ (((πΎ β π΄ β§ π β π΅ β§ π β π΅) β§ ππΆπ) β π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvrne.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | eqid 2732 | . . 3 β’ (ltβπΎ) = (ltβπΎ) | |
3 | cvrne.c | . . 3 β’ πΆ = ( β βπΎ) | |
4 | 1, 2, 3 | cvrlt 38135 | . 2 β’ (((πΎ β π΄ β§ π β π΅ β§ π β π΅) β§ ππΆπ) β π(ltβπΎ)π) |
5 | eqid 2732 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
6 | 5, 2 | pltval 18284 | . . 3 β’ ((πΎ β π΄ β§ π β π΅ β§ π β π΅) β (π(ltβπΎ)π β (π(leβπΎ)π β§ π β π))) |
7 | 6 | simplbda 500 | . 2 β’ (((πΎ β π΄ β§ π β π΅ β§ π β π΅) β§ π(ltβπΎ)π) β π β π) |
8 | 4, 7 | syldan 591 | 1 β’ (((πΎ β π΄ β§ π β π΅ β§ π β π΅) β§ ππΆπ) β π β π) |
Colors of variables: wff setvar class |
Syntax hints: 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 17143 lecple 17203 ltcplt 18260 β ccvr 38127 |
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-pow 5363 ax-pr 5427 ax-un 7724 |
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-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 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-plt 18282 df-covers 38131 |
This theorem is referenced by: cvrnrefN 38147 cvrcmp 38148 cdleme3b 39095 cdleme3c 39096 cdleme7e 39113 |
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