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Theorem lpolconN 40862
Description: Contraposition property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lpolcon.v 𝑉 = (Baseβ€˜π‘Š)
lpolcon.p 𝑃 = (LPolβ€˜π‘Š)
lpolcon.w (πœ‘ β†’ π‘Š ∈ 𝑋)
lpolcon.o (πœ‘ β†’ βŠ₯ ∈ 𝑃)
lpolcon.x (πœ‘ β†’ 𝑋 βŠ† 𝑉)
lpolcon.y (πœ‘ β†’ π‘Œ βŠ† 𝑉)
lpolcon.c (πœ‘ β†’ 𝑋 βŠ† π‘Œ)
Assertion
Ref Expression
lpolconN (πœ‘ β†’ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹))

Proof of Theorem lpolconN
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lpolcon.o . . 3 (πœ‘ β†’ βŠ₯ ∈ 𝑃)
2 lpolcon.w . . . 4 (πœ‘ β†’ π‘Š ∈ 𝑋)
3 lpolcon.v . . . . 5 𝑉 = (Baseβ€˜π‘Š)
4 eqid 2724 . . . . 5 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
5 eqid 2724 . . . . 5 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
6 eqid 2724 . . . . 5 (LSAtomsβ€˜π‘Š) = (LSAtomsβ€˜π‘Š)
7 eqid 2724 . . . . 5 (LSHypβ€˜π‘Š) = (LSHypβ€˜π‘Š)
8 lpolcon.p . . . . 5 𝑃 = (LPolβ€˜π‘Š)
93, 4, 5, 6, 7, 8islpolN 40858 . . . 4 (π‘Š ∈ 𝑋 β†’ ( βŠ₯ ∈ 𝑃 ↔ ( βŠ₯ :𝒫 π‘‰βŸΆ(LSubSpβ€˜π‘Š) ∧ (( βŠ₯ β€˜π‘‰) = {(0gβ€˜π‘Š)} ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ (LSAtomsβ€˜π‘Š)(( βŠ₯ β€˜π‘₯) ∈ (LSHypβ€˜π‘Š) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯)))))
102, 9syl 17 . . 3 (πœ‘ β†’ ( βŠ₯ ∈ 𝑃 ↔ ( βŠ₯ :𝒫 π‘‰βŸΆ(LSubSpβ€˜π‘Š) ∧ (( βŠ₯ β€˜π‘‰) = {(0gβ€˜π‘Š)} ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ (LSAtomsβ€˜π‘Š)(( βŠ₯ β€˜π‘₯) ∈ (LSHypβ€˜π‘Š) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯)))))
111, 10mpbid 231 . 2 (πœ‘ β†’ ( βŠ₯ :𝒫 π‘‰βŸΆ(LSubSpβ€˜π‘Š) ∧ (( βŠ₯ β€˜π‘‰) = {(0gβ€˜π‘Š)} ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ (LSAtomsβ€˜π‘Š)(( βŠ₯ β€˜π‘₯) ∈ (LSHypβ€˜π‘Š) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯))))
12 simpr2 1192 . . 3 (( βŠ₯ :𝒫 π‘‰βŸΆ(LSubSpβ€˜π‘Š) ∧ (( βŠ₯ β€˜π‘‰) = {(0gβ€˜π‘Š)} ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ (LSAtomsβ€˜π‘Š)(( βŠ₯ β€˜π‘₯) ∈ (LSHypβ€˜π‘Š) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯))) β†’ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)))
13 lpolcon.x . . . . 5 (πœ‘ β†’ 𝑋 βŠ† 𝑉)
14 lpolcon.y . . . . 5 (πœ‘ β†’ π‘Œ βŠ† 𝑉)
15 lpolcon.c . . . . 5 (πœ‘ β†’ 𝑋 βŠ† π‘Œ)
1613, 14, 153jca 1125 . . . 4 (πœ‘ β†’ (𝑋 βŠ† 𝑉 ∧ π‘Œ βŠ† 𝑉 ∧ 𝑋 βŠ† π‘Œ))
173fvexi 6896 . . . . . . 7 𝑉 ∈ V
1817elpw2 5336 . . . . . 6 (𝑋 ∈ 𝒫 𝑉 ↔ 𝑋 βŠ† 𝑉)
1913, 18sylibr 233 . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝒫 𝑉)
2017elpw2 5336 . . . . . 6 (π‘Œ ∈ 𝒫 𝑉 ↔ π‘Œ βŠ† 𝑉)
2114, 20sylibr 233 . . . . 5 (πœ‘ β†’ π‘Œ ∈ 𝒫 𝑉)
22 sseq1 4000 . . . . . . . . 9 (π‘₯ = 𝑋 β†’ (π‘₯ βŠ† 𝑉 ↔ 𝑋 βŠ† 𝑉))
23 biidd 262 . . . . . . . . 9 (π‘₯ = 𝑋 β†’ (𝑦 βŠ† 𝑉 ↔ 𝑦 βŠ† 𝑉))
24 sseq1 4000 . . . . . . . . 9 (π‘₯ = 𝑋 β†’ (π‘₯ βŠ† 𝑦 ↔ 𝑋 βŠ† 𝑦))
2522, 23, 243anbi123d 1432 . . . . . . . 8 (π‘₯ = 𝑋 β†’ ((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) ↔ (𝑋 βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ 𝑋 βŠ† 𝑦)))
26 fveq2 6882 . . . . . . . . 9 (π‘₯ = 𝑋 β†’ ( βŠ₯ β€˜π‘₯) = ( βŠ₯ β€˜π‘‹))
2726sseq2d 4007 . . . . . . . 8 (π‘₯ = 𝑋 β†’ (( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯) ↔ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘‹)))
2825, 27imbi12d 344 . . . . . . 7 (π‘₯ = 𝑋 β†’ (((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) ↔ ((𝑋 βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ 𝑋 βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘‹))))
29 biidd 262 . . . . . . . . 9 (𝑦 = π‘Œ β†’ (𝑋 βŠ† 𝑉 ↔ 𝑋 βŠ† 𝑉))
30 sseq1 4000 . . . . . . . . 9 (𝑦 = π‘Œ β†’ (𝑦 βŠ† 𝑉 ↔ π‘Œ βŠ† 𝑉))
31 sseq2 4001 . . . . . . . . 9 (𝑦 = π‘Œ β†’ (𝑋 βŠ† 𝑦 ↔ 𝑋 βŠ† π‘Œ))
3229, 30, 313anbi123d 1432 . . . . . . . 8 (𝑦 = π‘Œ β†’ ((𝑋 βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ 𝑋 βŠ† 𝑦) ↔ (𝑋 βŠ† 𝑉 ∧ π‘Œ βŠ† 𝑉 ∧ 𝑋 βŠ† π‘Œ)))
33 fveq2 6882 . . . . . . . . 9 (𝑦 = π‘Œ β†’ ( βŠ₯ β€˜π‘¦) = ( βŠ₯ β€˜π‘Œ))
3433sseq1d 4006 . . . . . . . 8 (𝑦 = π‘Œ β†’ (( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘‹) ↔ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹)))
3532, 34imbi12d 344 . . . . . . 7 (𝑦 = π‘Œ β†’ (((𝑋 βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ 𝑋 βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘‹)) ↔ ((𝑋 βŠ† 𝑉 ∧ π‘Œ βŠ† 𝑉 ∧ 𝑋 βŠ† π‘Œ) β†’ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹))))
3628, 35sylan9bb 509 . . . . . 6 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ (((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) ↔ ((𝑋 βŠ† 𝑉 ∧ π‘Œ βŠ† 𝑉 ∧ 𝑋 βŠ† π‘Œ) β†’ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹))))
3736spc2gv 3582 . . . . 5 ((𝑋 ∈ 𝒫 𝑉 ∧ π‘Œ ∈ 𝒫 𝑉) β†’ (βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) β†’ ((𝑋 βŠ† 𝑉 ∧ π‘Œ βŠ† 𝑉 ∧ 𝑋 βŠ† π‘Œ) β†’ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹))))
3819, 21, 37syl2anc 583 . . . 4 (πœ‘ β†’ (βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) β†’ ((𝑋 βŠ† 𝑉 ∧ π‘Œ βŠ† 𝑉 ∧ 𝑋 βŠ† π‘Œ) β†’ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹))))
3916, 38mpid 44 . . 3 (πœ‘ β†’ (βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) β†’ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹)))
4012, 39syl5 34 . 2 (πœ‘ β†’ (( βŠ₯ :𝒫 π‘‰βŸΆ(LSubSpβ€˜π‘Š) ∧ (( βŠ₯ β€˜π‘‰) = {(0gβ€˜π‘Š)} ∧ βˆ€π‘₯βˆ€π‘¦((π‘₯ βŠ† 𝑉 ∧ 𝑦 βŠ† 𝑉 ∧ π‘₯ βŠ† 𝑦) β†’ ( βŠ₯ β€˜π‘¦) βŠ† ( βŠ₯ β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ (LSAtomsβ€˜π‘Š)(( βŠ₯ β€˜π‘₯) ∈ (LSHypβ€˜π‘Š) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘₯)) = π‘₯))) β†’ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹)))
4111, 40mpd 15 1 (πœ‘ β†’ ( 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  βˆ€wal 1531   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053   βŠ† wss 3941  π’« cpw 4595  {csn 4621  βŸΆwf 6530  β€˜cfv 6534  Basecbs 17149  0gc0g 17390  LSubSpclss 20774  LSAtomsclsa 38348  LSHypclsh 38349  LPolclpoN 40855
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 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819  df-lpolN 40856
This theorem is referenced by: (None)
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