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Theorem frgrwopreglem3 28212
 Description: Lemma 3 for frgrwopreg 28221. 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 321 . . . 4 (𝑥 = 𝑌 → (¬ (𝐷𝑥) = 𝐾 ↔ ¬ (𝐷𝑌) = 𝐾))
3 frgrwopreg.b . . . . 5 𝐵 = (𝑉𝐴)
4 frgrwopreg.a . . . . . 6 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
54difeq2i 4027 . . . . 5 (𝑉𝐴) = (𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾})
6 notrab 4216 . . . . 5 (𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}) = {𝑥𝑉 ∣ ¬ (𝐷𝑥) = 𝐾}
73, 5, 63eqtri 2785 . . . 4 𝐵 = {𝑥𝑉 ∣ ¬ (𝐷𝑥) = 𝐾}
82, 7elrab2 3607 . . 3 (𝑌𝐵 ↔ (𝑌𝑉 ∧ ¬ (𝐷𝑌) = 𝐾))
9 fveqeq2 6672 . . . . . 6 (𝑥 = 𝑋 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑋) = 𝐾))
109, 4elrab2 3607 . . . . 5 (𝑋𝐴 ↔ (𝑋𝑉 ∧ (𝐷𝑋) = 𝐾))
11 eqeq2 2770 . . . . . . 7 ((𝐷𝑋) = 𝐾 → ((𝐷𝑌) = (𝐷𝑋) ↔ (𝐷𝑌) = 𝐾))
1211notbid 321 . . . . . 6 ((𝐷𝑋) = 𝐾 → (¬ (𝐷𝑌) = (𝐷𝑋) ↔ ¬ (𝐷𝑌) = 𝐾))
13 neqne 2959 . . . . . . 7 (¬ (𝐷𝑌) = (𝐷𝑋) → (𝐷𝑌) ≠ (𝐷𝑋))
1413necomd 3006 . . . . . 6 (¬ (𝐷𝑌) = (𝐷𝑋) → (𝐷𝑋) ≠ (𝐷𝑌))
1512, 14syl6bir 257 . . . . 5 ((𝐷𝑋) = 𝐾 → (¬ (𝐷𝑌) = 𝐾 → (𝐷𝑋) ≠ (𝐷𝑌)))
1610, 15simplbiim 508 . . . 4 (𝑋𝐴 → (¬ (𝐷𝑌) = 𝐾 → (𝐷𝑋) ≠ (𝐷𝑌)))
1716com12 32 . . 3 (¬ (𝐷𝑌) = 𝐾 → (𝑋𝐴 → (𝐷𝑋) ≠ (𝐷𝑌)))
188, 17simplbiim 508 . 2 (𝑌𝐵 → (𝑋𝐴 → (𝐷𝑋) ≠ (𝐷𝑌)))
1918impcom 411 1 ((𝑋𝐴𝑌𝐵) → (𝐷𝑋) ≠ (𝐷𝑌))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  {crab 3074   ∖ cdif 3857  ‘cfv 6340  Vtxcvtx 26902  VtxDegcvtxdg 27368 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-iota 6299  df-fv 6348 This theorem is referenced by:  frgrwopreglem4  28213  frgrwopreglem5lem  28218
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