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

Theorem frgrwopreglem3 28007
Description: Lemma 3 for frgrwopreg 28016. 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 6676 . . . . 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 4100 . . . . 5 (𝑉𝐴) = (𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾})
6 notrab 4284 . . . . 5 (𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}) = {𝑥𝑉 ∣ ¬ (𝐷𝑥) = 𝐾}
73, 5, 63eqtri 2853 . . . 4 𝐵 = {𝑥𝑉 ∣ ¬ (𝐷𝑥) = 𝐾}
82, 7elrab2 3687 . . 3 (𝑌𝐵 ↔ (𝑌𝑉 ∧ ¬ (𝐷𝑌) = 𝐾))
9 fveqeq2 6676 . . . . . 6 (𝑥 = 𝑋 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑋) = 𝐾))
109, 4elrab2 3687 . . . . 5 (𝑋𝐴 ↔ (𝑋𝑉 ∧ (𝐷𝑋) = 𝐾))
11 eqeq2 2838 . . . . . . 7 ((𝐷𝑋) = 𝐾 → ((𝐷𝑌) = (𝐷𝑋) ↔ (𝐷𝑌) = 𝐾))
1211notbid 319 . . . . . 6 ((𝐷𝑋) = 𝐾 → (¬ (𝐷𝑌) = (𝐷𝑋) ↔ ¬ (𝐷𝑌) = 𝐾))
13 neqne 3029 . . . . . . 7 (¬ (𝐷𝑌) = (𝐷𝑋) → (𝐷𝑌) ≠ (𝐷𝑋))
1413necomd 3076 . . . . . 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 1530  wcel 2107  wne 3021  {crab 3147  cdif 3937  cfv 6352  Vtxcvtx 26695  VtxDegcvtxdg 27161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-iota 6312  df-fv 6360
This theorem is referenced by:  frgrwopreglem4  28008  frgrwopreglem5lem  28013
  Copyright terms: Public domain W3C validator