Theorem frgrwopreglem3 28678

Description: Lemma 3 for frgrwopreg 28687. 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 6783	. . . . 5 ⊢ (𝑥 = 𝑌 → ((𝐷‘𝑥) = 𝐾 ↔ (𝐷‘𝑌) = 𝐾))
2	1	notbid 318	. . . 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 4054	. . . . 5 ⊢ (𝑉 ∖ 𝐴) = (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾})
6		notrab 4245	. . . . 5 ⊢ (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = {𝑥 ∈ 𝑉 ∣ ¬ (𝐷‘𝑥) = 𝐾}
7	3, 5, 6	3eqtri 2770	. . . 4 ⊢ 𝐵 = {𝑥 ∈ 𝑉 ∣ ¬ (𝐷‘𝑥) = 𝐾}
8	2, 7	elrab2 3627	. . 3 ⊢ (𝑌 ∈ 𝐵 ↔ (𝑌 ∈ 𝑉 ∧ ¬ (𝐷‘𝑌) = 𝐾))
9		fveqeq2 6783	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐷‘𝑥) = 𝐾 ↔ (𝐷‘𝑋) = 𝐾))
10	9, 4	elrab2 3627	. . . . 5 ⊢ (𝑋 ∈ 𝐴 ↔ (𝑋 ∈ 𝑉 ∧ (𝐷‘𝑋) = 𝐾))
11		eqeq2 2750	. . . . . . 7 ⊢ ((𝐷‘𝑋) = 𝐾 → ((𝐷‘𝑌) = (𝐷‘𝑋) ↔ (𝐷‘𝑌) = 𝐾))
12	11	notbid 318	. . . . . 6 ⊢ ((𝐷‘𝑋) = 𝐾 → (¬ (𝐷‘𝑌) = (𝐷‘𝑋) ↔ ¬ (𝐷‘𝑌) = 𝐾))
13		neqne 2951	. . . . . . 7 ⊢ (¬ (𝐷‘𝑌) = (𝐷‘𝑋) → (𝐷‘𝑌) ≠ (𝐷‘𝑋))
14	13	necomd 2999	. . . . . 6 ⊢ (¬ (𝐷‘𝑌) = (𝐷‘𝑋) → (𝐷‘𝑋) ≠ (𝐷‘𝑌))
15	12, 14	syl6bir 253	. . . . 5 ⊢ ((𝐷‘𝑋) = 𝐾 → (¬ (𝐷‘𝑌) = 𝐾 → (𝐷‘𝑋) ≠ (𝐷‘𝑌)))
16	10, 15	simplbiim 505	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (¬ (𝐷‘𝑌) = 𝐾 → (𝐷‘𝑋) ≠ (𝐷‘𝑌)))
17	16	com12 32	. . 3 ⊢ (¬ (𝐷‘𝑌) = 𝐾 → (𝑋 ∈ 𝐴 → (𝐷‘𝑋) ≠ (𝐷‘𝑌)))
18	8, 17	simplbiim 505	. 2 ⊢ (𝑌 ∈ 𝐵 → (𝑋 ∈ 𝐴 → (𝐷‘𝑋) ≠ (𝐷‘𝑌)))
19	18	impcom 408	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝐷‘𝑋) ≠ (𝐷‘𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  {crab 3068   ∖ cdif 3884  ‘cfv 6433  Vtxcvtx 27366  VtxDegcvtxdg 27832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441

This theorem is referenced by:  frgrwopreglem4  28679  frgrwopreglem5lem  28684