Theorem frgrwopreglem3 30942

Description: Lemma 3 for frgrwopreg 30951. 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 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
frgrwopreg.b	⊢ 𝐵 = (𝑉 ∖ 𝐴)

Assertion

Ref	Expression
frgrwopreglem3	⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝐷‘𝑋) ≠ (𝐷‘𝑌))

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

Step	Hyp	Ref	Expression
1		fveqeq2 6914	. . . . 5 ⊢ (𝑥 = 𝑌 → ((𝐷‘𝑥) = 𝐾 ↔ (𝐷‘𝑌) = 𝐾))
2	1	notbid 321	. . . 4 ⊢ (𝑥 = 𝑌 → (¬ (𝐷‘𝑥) = 𝐾 ↔ ¬ (𝐷‘𝑌) = 𝐾))
3		frgrwopreg.b	. . . . 5 ⊢ 𝐵 = (𝑉 ∖ 𝐴)
4		frgrwopreg.a	. . . . . 6 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
5	4	difeq2i 4063	. . . . 5 ⊢ (𝑉 ∖ 𝐴) = (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾})
6		notrab 4270	. . . . 5 ⊢ (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = {𝑥 ∈ 𝑉 ∣ ¬ (𝐷‘𝑥) = 𝐾}
7	3, 5, 6	3eqtri 2790	. . . 4 ⊢ 𝐵 = {𝑥 ∈ 𝑉 ∣ ¬ (𝐷‘𝑥) = 𝐾}
8	2, 7	elrab2 3652	. . 3 ⊢ (𝑌 ∈ 𝐵 ↔ (𝑌 ∈ 𝑉 ∧ ¬ (𝐷‘𝑌) = 𝐾))
9		fveqeq2 6914	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐷‘𝑥) = 𝐾 ↔ (𝐷‘𝑋) = 𝐾))
10	9, 4	elrab2 3652	. . . . 5 ⊢ (𝑋 ∈ 𝐴 ↔ (𝑋 ∈ 𝑉 ∧ (𝐷‘𝑋) = 𝐾))
11		eqeq2 2775	. . . . . . 7 ⊢ ((𝐷‘𝑋) = 𝐾 → ((𝐷‘𝑌) = (𝐷‘𝑋) ↔ (𝐷‘𝑌) = 𝐾))
12	11	notbid 321	. . . . . 6 ⊢ ((𝐷‘𝑋) = 𝐾 → (¬ (𝐷‘𝑌) = (𝐷‘𝑋) ↔ ¬ (𝐷‘𝑌) = 𝐾))
13		neqne 2967	. . . . . . 7 ⊢ (¬ (𝐷‘𝑌) = (𝐷‘𝑋) → (𝐷‘𝑌) ≠ (𝐷‘𝑋))
14	13	necomd 3014	. . . . . 6 ⊢ (¬ (𝐷‘𝑌) = (𝐷‘𝑋) → (𝐷‘𝑋) ≠ (𝐷‘𝑌))
15	12, 14	biimtrrdi 257	. . . . 5 ⊢ ((𝐷‘𝑋) = 𝐾 → (¬ (𝐷‘𝑌) = 𝐾 → (𝐷‘𝑋) ≠ (𝐷‘𝑌)))
16	10, 15	simplbiim 514	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (¬ (𝐷‘𝑌) = 𝐾 → (𝐷‘𝑋) ≠ (𝐷‘𝑌)))
17	16	com12 33	. . 3 ⊢ (¬ (𝐷‘𝑌) = 𝐾 → (𝑋 ∈ 𝐴 → (𝐷‘𝑋) ≠ (𝐷‘𝑌)))
18	8, 17	simplbiim 514	. 2 ⊢ (𝑌 ∈ 𝐵 → (𝑋 ∈ 𝐴 → (𝐷‘𝑋) ≠ (𝐷‘𝑌)))
19	18	impcom 413	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝐷‘𝑋) ≠ (𝐷‘𝑌))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  {crab 3416   ∖ cdif 4048  ‘cfv 6555  Vtxcvtx 29601  VtxDegcvtxdg 30073

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6509  df-fv 6563

This theorem is used by:  frgrwopreglem4  30943  frgrwopreglem5lem  30948