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Theorem 2polcon4bN 39302
Description: Contraposition law for polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2polss.a 𝐴 = (Atomsβ€˜πΎ)
2polss.p βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
Assertion
Ref Expression
2polcon4bN ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)) ↔ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹)))

Proof of Theorem 2polcon4bN
StepHypRef Expression
1 simpl1 1188 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))) β†’ 𝐾 ∈ HL)
2 simp1 1133 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ 𝐾 ∈ HL)
3 2polss.a . . . . . . . . 9 𝐴 = (Atomsβ€˜πΎ)
4 2polss.p . . . . . . . . 9 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
53, 4polssatN 39292 . . . . . . . 8 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘Œ) βŠ† 𝐴)
653adant2 1128 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘Œ) βŠ† 𝐴)
73, 4polssatN 39292 . . . . . . 7 ((𝐾 ∈ HL ∧ ( βŠ₯ β€˜π‘Œ) βŠ† 𝐴) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)) βŠ† 𝐴)
82, 6, 7syl2anc 583 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)) βŠ† 𝐴)
98adantr 480 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)) βŠ† 𝐴)
10 simpr 484 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)))
113, 4polcon3N 39301 . . . . 5 ((𝐾 ∈ HL ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)) βŠ† 𝐴 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))) βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜( βŠ₯ β€˜π‘‹))))
121, 9, 10, 11syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))) βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜( βŠ₯ β€˜π‘‹))))
1312ex 412 . . 3 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))) βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)))))
143, 43polN 39300 . . . . 5 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))) = ( βŠ₯ β€˜π‘Œ))
15143adant2 1128 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))) = ( βŠ₯ β€˜π‘Œ))
163, 43polN 39300 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜( βŠ₯ β€˜π‘‹))) = ( βŠ₯ β€˜π‘‹))
17163adant3 1129 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜( βŠ₯ β€˜π‘‹))) = ( βŠ₯ β€˜π‘‹))
1815, 17sseq12d 4010 . . 3 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (( βŠ₯ β€˜( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))) βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜( βŠ₯ β€˜π‘‹))) ↔ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹)))
1913, 18sylibd 238 . 2 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)) β†’ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹)))
20 simpl1 1188 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹)) β†’ 𝐾 ∈ HL)
213, 4polssatN 39292 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘‹) βŠ† 𝐴)
22213adant3 1129 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘‹) βŠ† 𝐴)
2322adantr 480 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹)) β†’ ( βŠ₯ β€˜π‘‹) βŠ† 𝐴)
24 simpr 484 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹)) β†’ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹))
253, 4polcon3N 39301 . . . 4 ((𝐾 ∈ HL ∧ ( βŠ₯ β€˜π‘‹) βŠ† 𝐴 ∧ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹)) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)))
2620, 23, 24, 25syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹)) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)))
2726ex 412 . 2 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))))
2819, 27impbid 211 1 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)) ↔ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943  β€˜cfv 6537  Atomscatm 38646  HLchlt 38733  βŠ₯𝑃cpolN 39286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-p1 18391  df-lat 18397  df-clat 18464  df-oposet 38559  df-ol 38561  df-oml 38562  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734  df-psubsp 38887  df-pmap 38888  df-polarityN 39287
This theorem is referenced by:  paddunN  39311
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