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Theorem frgrwopreglem3 27489
Description: Lemma 3 for frgrwopreg 27498. 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 fveq2 6353 . . . . . 6 (𝑥 = 𝑌 → (𝐷𝑥) = (𝐷𝑌))
21eqeq1d 2762 . . . . 5 (𝑥 = 𝑌 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑌) = 𝐾))
32notbid 307 . . . 4 (𝑥 = 𝑌 → (¬ (𝐷𝑥) = 𝐾 ↔ ¬ (𝐷𝑌) = 𝐾))
4 frgrwopreg.b . . . . 5 𝐵 = (𝑉𝐴)
5 frgrwopreg.a . . . . . 6 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
65difeq2i 3868 . . . . 5 (𝑉𝐴) = (𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾})
7 notrab 4047 . . . . 5 (𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}) = {𝑥𝑉 ∣ ¬ (𝐷𝑥) = 𝐾}
84, 6, 73eqtri 2786 . . . 4 𝐵 = {𝑥𝑉 ∣ ¬ (𝐷𝑥) = 𝐾}
93, 8elrab2 3507 . . 3 (𝑌𝐵 ↔ (𝑌𝑉 ∧ ¬ (𝐷𝑌) = 𝐾))
10 fveq2 6353 . . . . . . 7 (𝑥 = 𝑋 → (𝐷𝑥) = (𝐷𝑋))
1110eqeq1d 2762 . . . . . 6 (𝑥 = 𝑋 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑋) = 𝐾))
1211, 5elrab2 3507 . . . . 5 (𝑋𝐴 ↔ (𝑋𝑉 ∧ (𝐷𝑋) = 𝐾))
13 eqeq2 2771 . . . . . . 7 ((𝐷𝑋) = 𝐾 → ((𝐷𝑌) = (𝐷𝑋) ↔ (𝐷𝑌) = 𝐾))
1413notbid 307 . . . . . 6 ((𝐷𝑋) = 𝐾 → (¬ (𝐷𝑌) = (𝐷𝑋) ↔ ¬ (𝐷𝑌) = 𝐾))
15 neqne 2940 . . . . . . 7 (¬ (𝐷𝑌) = (𝐷𝑋) → (𝐷𝑌) ≠ (𝐷𝑋))
1615necomd 2987 . . . . . 6 (¬ (𝐷𝑌) = (𝐷𝑋) → (𝐷𝑋) ≠ (𝐷𝑌))
1714, 16syl6bir 244 . . . . 5 ((𝐷𝑋) = 𝐾 → (¬ (𝐷𝑌) = 𝐾 → (𝐷𝑋) ≠ (𝐷𝑌)))
1812, 17simplbiim 661 . . . 4 (𝑋𝐴 → (¬ (𝐷𝑌) = 𝐾 → (𝐷𝑋) ≠ (𝐷𝑌)))
1918com12 32 . . 3 (¬ (𝐷𝑌) = 𝐾 → (𝑋𝐴 → (𝐷𝑋) ≠ (𝐷𝑌)))
209, 19simplbiim 661 . 2 (𝑌𝐵 → (𝑋𝐴 → (𝐷𝑋) ≠ (𝐷𝑌)))
2120impcom 445 1 ((𝑋𝐴𝑌𝐵) → (𝐷𝑋) ≠ (𝐷𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wcel 2139  wne 2932  {crab 3054  cdif 3712  cfv 6049  Vtxcvtx 26094  VtxDegcvtxdg 26592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057
This theorem is referenced by:  frgrwopreglem4  27490  frgrwopreglem5lem  27495
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