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Theorem polcon3N 39092
Description: Contraposition law for polarity. Remark in [Holland95] p. 223. (Contributed by NM, 23-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
polcon3N ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† π‘Œ) β†’ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹))
Proof of Theorem polcon3N
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 simp3 1137 . . 3 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† π‘Œ) β†’ 𝑋 βŠ† π‘Œ)
2 iinss1 5012 . . 3 (𝑋 βŠ† π‘Œ β†’ ∩ 𝑝 ∈ π‘Œ ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘)) βŠ† ∩ 𝑝 ∈ 𝑋 ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘)))
3 sslin 4234 . . 3 (∩ 𝑝 ∈ π‘Œ ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘)) βŠ† ∩ 𝑝 ∈ 𝑋 ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘)) β†’ (𝐴 ∩ ∩ 𝑝 ∈ π‘Œ ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘))) βŠ† (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘))))
41, 2, 33syl 18 . 2 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† π‘Œ) β†’ (𝐴 ∩ ∩ 𝑝 ∈ π‘Œ ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘))) βŠ† (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘))))
5 eqid 2731 . . . 4 (ocβ€˜πΎ) = (ocβ€˜πΎ)
6 2polss.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
7 eqid 2731 . . . 4 (pmapβ€˜πΎ) = (pmapβ€˜πΎ)
8 2polss.p . . . 4 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
95, 6, 7, 8polvalN 39080 . . 3 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘Œ) = (𝐴 ∩ ∩ 𝑝 ∈ π‘Œ ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘))))
1093adant3 1131 . 2 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† π‘Œ) β†’ ( βŠ₯ β€˜π‘Œ) = (𝐴 ∩ ∩ 𝑝 ∈ π‘Œ ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘))))
11 simp1 1135 . . 3 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† π‘Œ) β†’ 𝐾 ∈ HL)
12 simp2 1136 . . . 4 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† π‘Œ) β†’ π‘Œ βŠ† 𝐴)
131, 12sstrd 3992 . . 3 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† π‘Œ) β†’ 𝑋 βŠ† 𝐴)
145, 6, 7, 8polvalN 39080 . . 3 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘‹) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘))))
1511, 13, 14syl2anc 583 . 2 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† π‘Œ) β†’ ( βŠ₯ β€˜π‘‹) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘))))
164, 10, 153sstr4d 4029 1 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† π‘Œ) β†’ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ∩ cin 3947   βŠ† wss 3948  βˆ© ciin 4998  β€˜cfv 6543  occoc 17210  Atomscatm 38437  HLchlt 38524  pmapcpmap 38672  βŠ₯𝑃cpolN 39077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  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-iun 4999  df-iin 5000  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-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-polarityN 39078
This theorem is referenced by:  2polcon4bN  39093  polcon2N  39094  paddunN  39102
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