MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frgrwopreglem3 Structured version   Visualization version   GIF version

Theorem frgrwopreglem3 29258
Description: Lemma 3 for frgrwopreg 29267. 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 6851 . . . . 5 (π‘₯ = π‘Œ β†’ ((π·β€˜π‘₯) = 𝐾 ↔ (π·β€˜π‘Œ) = 𝐾))
21notbid 317 . . . 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 4079 . . . . 5 (𝑉 βˆ– 𝐴) = (𝑉 βˆ– {π‘₯ ∈ 𝑉 ∣ (π·β€˜π‘₯) = 𝐾})
6 notrab 4271 . . . . 5 (𝑉 βˆ– {π‘₯ ∈ 𝑉 ∣ (π·β€˜π‘₯) = 𝐾}) = {π‘₯ ∈ 𝑉 ∣ Β¬ (π·β€˜π‘₯) = 𝐾}
73, 5, 63eqtri 2768 . . . 4 𝐡 = {π‘₯ ∈ 𝑉 ∣ Β¬ (π·β€˜π‘₯) = 𝐾}
82, 7elrab2 3648 . . 3 (π‘Œ ∈ 𝐡 ↔ (π‘Œ ∈ 𝑉 ∧ Β¬ (π·β€˜π‘Œ) = 𝐾))
9 fveqeq2 6851 . . . . . 6 (π‘₯ = 𝑋 β†’ ((π·β€˜π‘₯) = 𝐾 ↔ (π·β€˜π‘‹) = 𝐾))
109, 4elrab2 3648 . . . . 5 (𝑋 ∈ 𝐴 ↔ (𝑋 ∈ 𝑉 ∧ (π·β€˜π‘‹) = 𝐾))
11 eqeq2 2748 . . . . . . 7 ((π·β€˜π‘‹) = 𝐾 β†’ ((π·β€˜π‘Œ) = (π·β€˜π‘‹) ↔ (π·β€˜π‘Œ) = 𝐾))
1211notbid 317 . . . . . 6 ((π·β€˜π‘‹) = 𝐾 β†’ (Β¬ (π·β€˜π‘Œ) = (π·β€˜π‘‹) ↔ Β¬ (π·β€˜π‘Œ) = 𝐾))
13 neqne 2951 . . . . . . 7 (Β¬ (π·β€˜π‘Œ) = (π·β€˜π‘‹) β†’ (π·β€˜π‘Œ) β‰  (π·β€˜π‘‹))
1413necomd 2999 . . . . . 6 (Β¬ (π·β€˜π‘Œ) = (π·β€˜π‘‹) β†’ (π·β€˜π‘‹) β‰  (π·β€˜π‘Œ))
1512, 14syl6bir 253 . . . . 5 ((π·β€˜π‘‹) = 𝐾 β†’ (Β¬ (π·β€˜π‘Œ) = 𝐾 β†’ (π·β€˜π‘‹) β‰  (π·β€˜π‘Œ)))
1610, 15simplbiim 505 . . . 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 505 . 2 (π‘Œ ∈ 𝐡 β†’ (𝑋 ∈ 𝐴 β†’ (π·β€˜π‘‹) β‰  (π·β€˜π‘Œ)))
1918impcom 408 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 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2943  {crab 3407   βˆ– cdif 3907  β€˜cfv 6496  Vtxcvtx 27947  VtxDegcvtxdg 28413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504
This theorem is referenced by:  frgrwopreglem4  29259  frgrwopreglem5lem  29264
  Copyright terms: Public domain W3C validator