Theorem frgrwopreglem3 41483

Description: Lemma 3 for frgrwopreg 41486. 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.)

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		fveq2 6103	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐷‘𝑥) = (𝐷‘𝑋))
2	1	eqeq1d 2612	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐷‘𝑥) = 𝐾 ↔ (𝐷‘𝑋) = 𝐾))
3		frgrwopreg.a	. . . 4 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
4	2, 3	elrab2 3333	. . 3 ⊢ (𝑋 ∈ 𝐴 ↔ (𝑋 ∈ 𝑉 ∧ (𝐷‘𝑋) = 𝐾))
5		frgrwopreg.b	. . . . . 6 ⊢ 𝐵 = (𝑉 ∖ 𝐴)
6	5	eleq2i 2680	. . . . 5 ⊢ (𝑌 ∈ 𝐵 ↔ 𝑌 ∈ (𝑉 ∖ 𝐴))
7		eldif 3550	. . . . 5 ⊢ (𝑌 ∈ (𝑉 ∖ 𝐴) ↔ (𝑌 ∈ 𝑉 ∧ ¬ 𝑌 ∈ 𝐴))
8	6, 7	bitri 263	. . . 4 ⊢ (𝑌 ∈ 𝐵 ↔ (𝑌 ∈ 𝑉 ∧ ¬ 𝑌 ∈ 𝐴))
9		fveq2 6103	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → (𝐷‘𝑥) = (𝐷‘𝑌))
10	9	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → ((𝐷‘𝑥) = 𝐾 ↔ (𝐷‘𝑌) = 𝐾))
11	10, 3	elrab2 3333	. . . . . . 7 ⊢ (𝑌 ∈ 𝐴 ↔ (𝑌 ∈ 𝑉 ∧ (𝐷‘𝑌) = 𝐾))
12		ianor 508	. . . . . . . 8 ⊢ (¬ (𝑌 ∈ 𝑉 ∧ (𝐷‘𝑌) = 𝐾) ↔ (¬ 𝑌 ∈ 𝑉 ∨ ¬ (𝐷‘𝑌) = 𝐾))
13		pm2.21 119	. . . . . . . . 9 ⊢ (¬ 𝑌 ∈ 𝑉 → (𝑌 ∈ 𝑉 → ((𝑋 ∈ 𝑉 ∧ (𝐷‘𝑋) = 𝐾) → (𝐷‘𝑋) ≠ (𝐷‘𝑌))))
14		neqne 2790	. . . . . . . . . . . . . 14 ⊢ (¬ (𝐷‘𝑌) = 𝐾 → (𝐷‘𝑌) ≠ 𝐾)
15	14	adantr 480	. . . . . . . . . . . . 13 ⊢ ((¬ (𝐷‘𝑌) = 𝐾 ∧ (𝑋 ∈ 𝑉 ∧ (𝐷‘𝑋) = 𝐾)) → (𝐷‘𝑌) ≠ 𝐾)
16	15	necomd 2837	. . . . . . . . . . . 12 ⊢ ((¬ (𝐷‘𝑌) = 𝐾 ∧ (𝑋 ∈ 𝑉 ∧ (𝐷‘𝑋) = 𝐾)) → 𝐾 ≠ (𝐷‘𝑌))
17		neeq1 2844	. . . . . . . . . . . . 13 ⊢ ((𝐷‘𝑋) = 𝐾 → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) ↔ 𝐾 ≠ (𝐷‘𝑌)))
18	17	ad2antll 761	. . . . . . . . . . . 12 ⊢ ((¬ (𝐷‘𝑌) = 𝐾 ∧ (𝑋 ∈ 𝑉 ∧ (𝐷‘𝑋) = 𝐾)) → ((𝐷‘𝑋) ≠ (𝐷‘𝑌) ↔ 𝐾 ≠ (𝐷‘𝑌)))
19	16, 18	mpbird 246	. . . . . . . . . . 11 ⊢ ((¬ (𝐷‘𝑌) = 𝐾 ∧ (𝑋 ∈ 𝑉 ∧ (𝐷‘𝑋) = 𝐾)) → (𝐷‘𝑋) ≠ (𝐷‘𝑌))
20	19	ex 449	. . . . . . . . . 10 ⊢ (¬ (𝐷‘𝑌) = 𝐾 → ((𝑋 ∈ 𝑉 ∧ (𝐷‘𝑋) = 𝐾) → (𝐷‘𝑋) ≠ (𝐷‘𝑌)))
21	20	a1d 25	. . . . . . . . 9 ⊢ (¬ (𝐷‘𝑌) = 𝐾 → (𝑌 ∈ 𝑉 → ((𝑋 ∈ 𝑉 ∧ (𝐷‘𝑋) = 𝐾) → (𝐷‘𝑋) ≠ (𝐷‘𝑌))))
22	13, 21	jaoi 393	. . . . . . . 8 ⊢ ((¬ 𝑌 ∈ 𝑉 ∨ ¬ (𝐷‘𝑌) = 𝐾) → (𝑌 ∈ 𝑉 → ((𝑋 ∈ 𝑉 ∧ (𝐷‘𝑋) = 𝐾) → (𝐷‘𝑋) ≠ (𝐷‘𝑌))))
23	12, 22	sylbi 206	. . . . . . 7 ⊢ (¬ (𝑌 ∈ 𝑉 ∧ (𝐷‘𝑌) = 𝐾) → (𝑌 ∈ 𝑉 → ((𝑋 ∈ 𝑉 ∧ (𝐷‘𝑋) = 𝐾) → (𝐷‘𝑋) ≠ (𝐷‘𝑌))))
24	11, 23	sylnbi 319	. . . . . 6 ⊢ (¬ 𝑌 ∈ 𝐴 → (𝑌 ∈ 𝑉 → ((𝑋 ∈ 𝑉 ∧ (𝐷‘𝑋) = 𝐾) → (𝐷‘𝑋) ≠ (𝐷‘𝑌))))
25	24	impcom 445	. . . . 5 ⊢ ((𝑌 ∈ 𝑉 ∧ ¬ 𝑌 ∈ 𝐴) → ((𝑋 ∈ 𝑉 ∧ (𝐷‘𝑋) = 𝐾) → (𝐷‘𝑋) ≠ (𝐷‘𝑌)))
26	25	com12 32	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐷‘𝑋) = 𝐾) → ((𝑌 ∈ 𝑉 ∧ ¬ 𝑌 ∈ 𝐴) → (𝐷‘𝑋) ≠ (𝐷‘𝑌)))
27	8, 26	syl5bi 231	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐷‘𝑋) = 𝐾) → (𝑌 ∈ 𝐵 → (𝐷‘𝑋) ≠ (𝐷‘𝑌)))
28	4, 27	sylbi 206	. 2 ⊢ (𝑋 ∈ 𝐴 → (𝑌 ∈ 𝐵 → (𝐷‘𝑋) ≠ (𝐷‘𝑌)))
29	28	imp 444	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝐷‘𝑋) ≠ (𝐷‘𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900   ∖ cdif 3537  ‘cfv 5804  Vtxcvtx 25673  VtxDegcvtxdg 40681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812

This theorem is referenced by:  frgrwopreglem4  41484