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Theorem frgrwopreglem3 29567
Description: Lemma 3 for frgrwopreg 29576. The vertices in the sets 𝐴 and 𝐡 have different degrees. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 2-Jan-2022.)
Hypotheses
Ref Expression
frgrwopreg.v 𝑉 = (Vtxβ€˜πΊ)
frgrwopreg.d 𝐷 = (VtxDegβ€˜πΊ)
frgrwopreg.a 𝐴 = {π‘₯ ∈ 𝑉 ∣ (π·β€˜π‘₯) = 𝐾}
frgrwopreg.b 𝐡 = (𝑉 βˆ– 𝐴)
Assertion
Ref Expression
frgrwopreglem3 ((𝑋 ∈ 𝐴 ∧ π‘Œ ∈ 𝐡) β†’ (π·β€˜π‘‹) β‰  (π·β€˜π‘Œ))
Distinct variable groups:   π‘₯,𝑉   π‘₯,𝐴   π‘₯,𝐺   π‘₯,𝐾   π‘₯,𝐷   π‘₯,𝑋   π‘₯,π‘Œ
Allowed substitution hint:   𝐡(π‘₯)
Proof of Theorem frgrwopreglem3
StepHypRef Expression
1 fveqeq2 6901 . . . . 5 (π‘₯ = π‘Œ β†’ ((π·β€˜π‘₯) = 𝐾 ↔ (π·β€˜π‘Œ) = 𝐾))
21notbid 318 . . . 4 (π‘₯ = π‘Œ β†’ (Β¬ (π·β€˜π‘₯) = 𝐾 ↔ Β¬ (π·β€˜π‘Œ) = 𝐾))
3 frgrwopreg.b . . . . 5 𝐡 = (𝑉 βˆ– 𝐴)
4 frgrwopreg.a . . . . . 6 𝐴 = {π‘₯ ∈ 𝑉 ∣ (π·β€˜π‘₯) = 𝐾}
54difeq2i 4120 . . . . 5 (𝑉 βˆ– 𝐴) = (𝑉 βˆ– {π‘₯ ∈ 𝑉 ∣ (π·β€˜π‘₯) = 𝐾})
6 notrab 4312 . . . . 5 (𝑉 βˆ– {π‘₯ ∈ 𝑉 ∣ (π·β€˜π‘₯) = 𝐾}) = {π‘₯ ∈ 𝑉 ∣ Β¬ (π·β€˜π‘₯) = 𝐾}
73, 5, 63eqtri 2765 . . . 4 𝐡 = {π‘₯ ∈ 𝑉 ∣ Β¬ (π·β€˜π‘₯) = 𝐾}
82, 7elrab2 3687 . . 3 (π‘Œ ∈ 𝐡 ↔ (π‘Œ ∈ 𝑉 ∧ Β¬ (π·β€˜π‘Œ) = 𝐾))
9 fveqeq2 6901 . . . . . 6 (π‘₯ = 𝑋 β†’ ((π·β€˜π‘₯) = 𝐾 ↔ (π·β€˜π‘‹) = 𝐾))
109, 4elrab2 3687 . . . . 5 (𝑋 ∈ 𝐴 ↔ (𝑋 ∈ 𝑉 ∧ (π·β€˜π‘‹) = 𝐾))
11 eqeq2 2745 . . . . . . 7 ((π·β€˜π‘‹) = 𝐾 β†’ ((π·β€˜π‘Œ) = (π·β€˜π‘‹) ↔ (π·β€˜π‘Œ) = 𝐾))
1211notbid 318 . . . . . 6 ((π·β€˜π‘‹) = 𝐾 β†’ (Β¬ (π·β€˜π‘Œ) = (π·β€˜π‘‹) ↔ Β¬ (π·β€˜π‘Œ) = 𝐾))
13 neqne 2949 . . . . . . 7 (Β¬ (π·β€˜π‘Œ) = (π·β€˜π‘‹) β†’ (π·β€˜π‘Œ) β‰  (π·β€˜π‘‹))
1413necomd 2997 . . . . . 6 (Β¬ (π·β€˜π‘Œ) = (π·β€˜π‘‹) β†’ (π·β€˜π‘‹) β‰  (π·β€˜π‘Œ))
1512, 14syl6bir 254 . . . . 5 ((π·β€˜π‘‹) = 𝐾 β†’ (Β¬ (π·β€˜π‘Œ) = 𝐾 β†’ (π·β€˜π‘‹) β‰  (π·β€˜π‘Œ)))
1610, 15simplbiim 506 . . . 4 (𝑋 ∈ 𝐴 β†’ (Β¬ (π·β€˜π‘Œ) = 𝐾 β†’ (π·β€˜π‘‹) β‰  (π·β€˜π‘Œ)))
1716com12 32 . . 3 (Β¬ (π·β€˜π‘Œ) = 𝐾 β†’ (𝑋 ∈ 𝐴 β†’ (π·β€˜π‘‹) β‰  (π·β€˜π‘Œ)))
188, 17simplbiim 506 . 2 (π‘Œ ∈ 𝐡 β†’ (𝑋 ∈ 𝐴 β†’ (π·β€˜π‘‹) β‰  (π·β€˜π‘Œ)))
1918impcom 409 1 ((𝑋 ∈ 𝐴 ∧ π‘Œ ∈ 𝐡) β†’ (π·β€˜π‘‹) β‰  (π·β€˜π‘Œ))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  {crab 3433   βˆ– cdif 3946  β€˜cfv 6544  Vtxcvtx 28256  VtxDegcvtxdg 28722
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-12 2172  ax-ext 2704
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-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552
This theorem is referenced by:  frgrwopreglem4  29568  frgrwopreglem5lem  29573
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