Theorem frgrawopreglem3 26573

Description: Lemma 3 for frgrawopreg 26576. The vertices in the sets A and B have different degrees. (Contributed by Alexander van der Vekens, 30-Dec-2017.)

Hypotheses

Ref	Expression
frgrawopreg.a	⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
frgrawopreg.b	⊢ 𝐵 = (𝑉 ∖ 𝐴)

Assertion

Ref	Expression
frgrawopreglem3	⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐸   𝑥,𝐾   𝑥,𝑉   𝑥,𝑋   𝑥,𝑌

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem frgrawopreglem3

Step	Hyp	Ref	Expression
1		fveq2 6103	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘𝑋))
2	1	eqeq1d 2612	. . . 4 ⊢ (𝑥 = 𝑋 → (((𝑉 VDeg 𝐸)‘𝑥) = 𝐾 ↔ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾))
3		frgrawopreg.a	. . . 4 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
4	2, 3	elrab2 3333	. . 3 ⊢ (𝑋 ∈ 𝐴 ↔ (𝑋 ∈ 𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾))
5		frgrawopreg.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 → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘𝑌))
10	9	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → (((𝑉 VDeg 𝐸)‘𝑥) = 𝐾 ↔ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾))
11	10, 3	elrab2 3333	. . . . . . 7 ⊢ (𝑌 ∈ 𝐴 ↔ (𝑌 ∈ 𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾))
12		ianor 508	. . . . . . . 8 ⊢ (¬ (𝑌 ∈ 𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾) ↔ (¬ 𝑌 ∈ 𝑉 ∨ ¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾))
13		pm2.21 119	. . . . . . . . 9 ⊢ (¬ 𝑌 ∈ 𝑉 → (𝑌 ∈ 𝑉 → ((𝑋 ∈ 𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))))
14		nesym 2838	. . . . . . . . . . . . . 14 ⊢ (𝐾 ≠ ((𝑉 VDeg 𝐸)‘𝑌) ↔ ¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾)
15	14	biimpri 217	. . . . . . . . . . . . 13 ⊢ (¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾 → 𝐾 ≠ ((𝑉 VDeg 𝐸)‘𝑌))
16		neeq1 2844	. . . . . . . . . . . . 13 ⊢ (((𝑉 VDeg 𝐸)‘𝑋) = 𝐾 → (((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌) ↔ 𝐾 ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
17	15, 16	syl5ibr 235	. . . . . . . . . . . 12 ⊢ (((𝑉 VDeg 𝐸)‘𝑋) = 𝐾 → (¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾 → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
18	17	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → (¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾 → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
19	18	com12 32	. . . . . . . . . 10 ⊢ (¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾 → ((𝑋 ∈ 𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
20	19	a1d 25	. . . . . . . . 9 ⊢ (¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾 → (𝑌 ∈ 𝑉 → ((𝑋 ∈ 𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))))
21	13, 20	jaoi 393	. . . . . . . 8 ⊢ ((¬ 𝑌 ∈ 𝑉 ∨ ¬ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾) → (𝑌 ∈ 𝑉 → ((𝑋 ∈ 𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))))
22	12, 21	sylbi 206	. . . . . . 7 ⊢ (¬ (𝑌 ∈ 𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑌) = 𝐾) → (𝑌 ∈ 𝑉 → ((𝑋 ∈ 𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))))
23	11, 22	sylnbi 319	. . . . . 6 ⊢ (¬ 𝑌 ∈ 𝐴 → (𝑌 ∈ 𝑉 → ((𝑋 ∈ 𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))))
24	23	impcom 445	. . . . 5 ⊢ ((𝑌 ∈ 𝑉 ∧ ¬ 𝑌 ∈ 𝐴) → ((𝑋 ∈ 𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
25	24	com12 32	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → ((𝑌 ∈ 𝑉 ∧ ¬ 𝑌 ∈ 𝐴) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
26	8, 25	syl5bi 231	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑋) = 𝐾) → (𝑌 ∈ 𝐵 → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
27	4, 26	sylbi 206	. 2 ⊢ (𝑋 ∈ 𝐴 → (𝑌 ∈ 𝐵 → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌)))
28	27	imp 444	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900   ∖ cdif 3537  ‘cfv 5804  (class class class)co 6549   VDeg cvdg 26420

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:  frgrawopreglem4  26574