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Theorem frgrwopreglem3 27594
Description: Lemma 3 for frgrwopreg 27603. 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 6384 . . . . 5 (𝑥 = 𝑌 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑌) = 𝐾))
21notbid 309 . . . 4 (𝑥 = 𝑌 → (¬ (𝐷𝑥) = 𝐾 ↔ ¬ (𝐷𝑌) = 𝐾))
3 frgrwopreg.b . . . . 5 𝐵 = (𝑉𝐴)
4 frgrwopreg.a . . . . . 6 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
54difeq2i 3887 . . . . 5 (𝑉𝐴) = (𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾})
6 notrab 4068 . . . . 5 (𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}) = {𝑥𝑉 ∣ ¬ (𝐷𝑥) = 𝐾}
73, 5, 63eqtri 2791 . . . 4 𝐵 = {𝑥𝑉 ∣ ¬ (𝐷𝑥) = 𝐾}
82, 7elrab2 3523 . . 3 (𝑌𝐵 ↔ (𝑌𝑉 ∧ ¬ (𝐷𝑌) = 𝐾))
9 fveqeq2 6384 . . . . . 6 (𝑥 = 𝑋 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑋) = 𝐾))
109, 4elrab2 3523 . . . . 5 (𝑋𝐴 ↔ (𝑋𝑉 ∧ (𝐷𝑋) = 𝐾))
11 eqeq2 2776 . . . . . . 7 ((𝐷𝑋) = 𝐾 → ((𝐷𝑌) = (𝐷𝑋) ↔ (𝐷𝑌) = 𝐾))
1211notbid 309 . . . . . 6 ((𝐷𝑋) = 𝐾 → (¬ (𝐷𝑌) = (𝐷𝑋) ↔ ¬ (𝐷𝑌) = 𝐾))
13 neqne 2945 . . . . . . 7 (¬ (𝐷𝑌) = (𝐷𝑋) → (𝐷𝑌) ≠ (𝐷𝑋))
1413necomd 2992 . . . . . 6 (¬ (𝐷𝑌) = (𝐷𝑋) → (𝐷𝑋) ≠ (𝐷𝑌))
1512, 14syl6bir 245 . . . . 5 ((𝐷𝑋) = 𝐾 → (¬ (𝐷𝑌) = 𝐾 → (𝐷𝑋) ≠ (𝐷𝑌)))
1610, 15simplbiim 499 . . . 4 (𝑋𝐴 → (¬ (𝐷𝑌) = 𝐾 → (𝐷𝑋) ≠ (𝐷𝑌)))
1716com12 32 . . 3 (¬ (𝐷𝑌) = 𝐾 → (𝑋𝐴 → (𝐷𝑋) ≠ (𝐷𝑌)))
188, 17simplbiim 499 . 2 (𝑌𝐵 → (𝑋𝐴 → (𝐷𝑋) ≠ (𝐷𝑌)))
1918impcom 396 1 ((𝑋𝐴𝑌𝐵) → (𝐷𝑋) ≠ (𝐷𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1652  wcel 2155  wne 2937  {crab 3059  cdif 3729  cfv 6068  Vtxcvtx 26165  VtxDegcvtxdg 26652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-iota 6031  df-fv 6076
This theorem is referenced by:  frgrwopreglem4  27595  frgrwopreglem5lem  27600
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