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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvrletrN | Structured version Visualization version GIF version |
Description: Property of an element above a covering. (Contributed by NM, 7-Dec-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cvrletr.b | β’ π΅ = (BaseβπΎ) |
cvrletr.l | β’ β€ = (leβπΎ) |
cvrletr.s | β’ < = (ltβπΎ) |
cvrletr.c | β’ πΆ = ( β βπΎ) |
Ref | Expression |
---|---|
cvrletrN | β’ ((πΎ β Poset β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((ππΆπ β§ π β€ π) β π < π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 766 | . . . 4 β’ (((πΎ β Poset β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β§ ππΆπ) β πΎ β Poset) | |
2 | simplr1 1216 | . . . 4 β’ (((πΎ β Poset β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β§ ππΆπ) β π β π΅) | |
3 | simplr2 1217 | . . . 4 β’ (((πΎ β Poset β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β§ ππΆπ) β π β π΅) | |
4 | simpr 486 | . . . 4 β’ (((πΎ β Poset β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β§ ππΆπ) β ππΆπ) | |
5 | cvrletr.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
6 | cvrletr.s | . . . . 5 β’ < = (ltβπΎ) | |
7 | cvrletr.c | . . . . 5 β’ πΆ = ( β βπΎ) | |
8 | 5, 6, 7 | cvrlt 38140 | . . . 4 β’ (((πΎ β Poset β§ π β π΅ β§ π β π΅) β§ ππΆπ) β π < π) |
9 | 1, 2, 3, 4, 8 | syl31anc 1374 | . . 3 β’ (((πΎ β Poset β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β§ ππΆπ) β π < π) |
10 | cvrletr.l | . . . . 5 β’ β€ = (leβπΎ) | |
11 | 5, 10, 6 | pltletr 18296 | . . . 4 β’ ((πΎ β Poset β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π < π β§ π β€ π) β π < π)) |
12 | 11 | adantr 482 | . . 3 β’ (((πΎ β Poset β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β§ ππΆπ) β ((π < π β§ π β€ π) β π < π)) |
13 | 9, 12 | mpand 694 | . 2 β’ (((πΎ β Poset β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β§ ππΆπ) β (π β€ π β π < π)) |
14 | 13 | expimpd 455 | 1 β’ ((πΎ β Poset β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((ππΆπ β§ π β€ π) β π < π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 class class class wbr 5149 βcfv 6544 Basecbs 17144 lecple 17204 Posetcpo 18260 ltcplt 18261 β ccvr 38132 |
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 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
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 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-proset 18248 df-poset 18266 df-plt 18283 df-covers 38136 |
This theorem is referenced by: (None) |
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