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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > polcon2bN | Structured version Visualization version GIF version |
Description: Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2polss.a | β’ π΄ = (AtomsβπΎ) |
2polss.p | β’ β₯ = (β₯πβπΎ) |
Ref | Expression |
---|---|
polcon2bN | β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β ( β₯ βπ) β π β ( β₯ βπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1192 | . . 3 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ π β ( β₯ βπ)) β πΎ β HL) | |
2 | simpl3 1194 | . . 3 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ π β ( β₯ βπ)) β π β π΄) | |
3 | simpr 486 | . . 3 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ π β ( β₯ βπ)) β π β ( β₯ βπ)) | |
4 | 2polss.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
5 | 2polss.p | . . . 4 β’ β₯ = (β₯πβπΎ) | |
6 | 4, 5 | polcon2N 38385 | . . 3 β’ ((πΎ β HL β§ π β π΄ β§ π β ( β₯ βπ)) β π β ( β₯ βπ)) |
7 | 1, 2, 3, 6 | syl3anc 1372 | . 2 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ π β ( β₯ βπ)) β π β ( β₯ βπ)) |
8 | simpl1 1192 | . . 3 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ π β ( β₯ βπ)) β πΎ β HL) | |
9 | simpl2 1193 | . . 3 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ π β ( β₯ βπ)) β π β π΄) | |
10 | simpr 486 | . . 3 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ π β ( β₯ βπ)) β π β ( β₯ βπ)) | |
11 | 4, 5 | polcon2N 38385 | . . 3 β’ ((πΎ β HL β§ π β π΄ β§ π β ( β₯ βπ)) β π β ( β₯ βπ)) |
12 | 8, 9, 10, 11 | syl3anc 1372 | . 2 β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ π β ( β₯ βπ)) β π β ( β₯ βπ)) |
13 | 7, 12 | impbida 800 | 1 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β ( β₯ βπ) β π β ( β₯ βπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wss 3911 βcfv 6497 Atomscatm 37728 HLchlt 37815 β₯πcpolN 38368 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-iin 4958 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-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-proset 18185 df-poset 18203 df-plt 18220 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 df-p1 18316 df-lat 18322 df-clat 18389 df-oposet 37641 df-ol 37643 df-oml 37644 df-covers 37731 df-ats 37732 df-atl 37763 df-cvlat 37787 df-hlat 37816 df-psubsp 37969 df-pmap 37970 df-polarityN 38369 |
This theorem is referenced by: (None) |
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