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Theorem frgrwopreglem3 28020
Description: Lemma 3 for frgrwopreg 28029. The vertices in the sets 𝐴 and 𝐵 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 = (Vtx‘𝐺)
frgrwopreg.d 𝐷 = (VtxDeg‘𝐺)
frgrwopreg.a 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
frgrwopreg.b 𝐵 = (𝑉𝐴)
Assertion
Ref Expression
frgrwopreglem3 ((𝑋𝐴𝑌𝐵) → (𝐷𝑋) ≠ (𝐷𝑌))
Distinct variable groups:   𝑥,𝑉   𝑥,𝐴   𝑥,𝐺   𝑥,𝐾   𝑥,𝐷   𝑥,𝑋   𝑥,𝑌
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem frgrwopreglem3
StepHypRef Expression
1 fveqeq2 6672 . . . . 5 (𝑥 = 𝑌 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑌) = 𝐾))
21notbid 319 . . . 4 (𝑥 = 𝑌 → (¬ (𝐷𝑥) = 𝐾 ↔ ¬ (𝐷𝑌) = 𝐾))
3 frgrwopreg.b . . . . 5 𝐵 = (𝑉𝐴)
4 frgrwopreg.a . . . . . 6 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
54difeq2i 4093 . . . . 5 (𝑉𝐴) = (𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾})
6 notrab 4277 . . . . 5 (𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}) = {𝑥𝑉 ∣ ¬ (𝐷𝑥) = 𝐾}
73, 5, 63eqtri 2845 . . . 4 𝐵 = {𝑥𝑉 ∣ ¬ (𝐷𝑥) = 𝐾}
82, 7elrab2 3680 . . 3 (𝑌𝐵 ↔ (𝑌𝑉 ∧ ¬ (𝐷𝑌) = 𝐾))
9 fveqeq2 6672 . . . . . 6 (𝑥 = 𝑋 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑋) = 𝐾))
109, 4elrab2 3680 . . . . 5 (𝑋𝐴 ↔ (𝑋𝑉 ∧ (𝐷𝑋) = 𝐾))
11 eqeq2 2830 . . . . . . 7 ((𝐷𝑋) = 𝐾 → ((𝐷𝑌) = (𝐷𝑋) ↔ (𝐷𝑌) = 𝐾))
1211notbid 319 . . . . . 6 ((𝐷𝑋) = 𝐾 → (¬ (𝐷𝑌) = (𝐷𝑋) ↔ ¬ (𝐷𝑌) = 𝐾))
13 neqne 3021 . . . . . . 7 (¬ (𝐷𝑌) = (𝐷𝑋) → (𝐷𝑌) ≠ (𝐷𝑋))
1413necomd 3068 . . . . . 6 (¬ (𝐷𝑌) = (𝐷𝑋) → (𝐷𝑋) ≠ (𝐷𝑌))
1512, 14syl6bir 255 . . . . 5 ((𝐷𝑋) = 𝐾 → (¬ (𝐷𝑌) = 𝐾 → (𝐷𝑋) ≠ (𝐷𝑌)))
1610, 15simplbiim 505 . . . 4 (𝑋𝐴 → (¬ (𝐷𝑌) = 𝐾 → (𝐷𝑋) ≠ (𝐷𝑌)))
1716com12 32 . . 3 (¬ (𝐷𝑌) = 𝐾 → (𝑋𝐴 → (𝐷𝑋) ≠ (𝐷𝑌)))
188, 17simplbiim 505 . 2 (𝑌𝐵 → (𝑋𝐴 → (𝐷𝑋) ≠ (𝐷𝑌)))
1918impcom 408 1 ((𝑋𝐴𝑌𝐵) → (𝐷𝑋) ≠ (𝐷𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  wne 3013  {crab 3139  cdif 3930  cfv 6348  Vtxcvtx 26708  VtxDegcvtxdg 27174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356
This theorem is referenced by:  frgrwopreglem4  28021  frgrwopreglem5lem  28026
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