Theorem uhgr3cyclexlem 30437

Description: Lemma for uhgr3cyclex 30438. (Contributed by AV, 12-Feb-2021.)

Hypotheses

Ref	Expression
uhgr3cyclex.v	⊢ 𝑉 = (Vtx‘𝐺)
uhgr3cyclex.e	⊢ 𝐸 = (Edg‘𝐺)
uhgr3cyclex.i	⊢ 𝐼 = (iEdg‘𝐺)

Assertion

Ref	Expression
uhgr3cyclexlem	⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼‘𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼‘𝐾)))) → 𝐽 ≠ 𝐾)

Proof of Theorem uhgr3cyclexlem

Step	Hyp	Ref	Expression
1		fveq2 6871	. . . . . . . . 9 ⊢ (𝐽 = 𝐾 → (𝐼‘𝐽) = (𝐼‘𝐾))
2	1	eqeq2d 2776	. . . . . . . 8 ⊢ (𝐽 = 𝐾 → ({𝐵, 𝐶} = (𝐼‘𝐽) ↔ {𝐵, 𝐶} = (𝐼‘𝐾)))
3		eqeq2 2777	. . . . . . . . . . . 12 ⊢ ((𝐼‘𝐾) = {𝐶, 𝐴} → ({𝐵, 𝐶} = (𝐼‘𝐾) ↔ {𝐵, 𝐶} = {𝐶, 𝐴}))
4	3	eqcoms 2773	. . . . . . . . . . 11 ⊢ ({𝐶, 𝐴} = (𝐼‘𝐾) → ({𝐵, 𝐶} = (𝐼‘𝐾) ↔ {𝐵, 𝐶} = {𝐶, 𝐴}))
5		prcom 4694	. . . . . . . . . . . . . 14 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
6	5	eqeq1i 2770	. . . . . . . . . . . . 13 ⊢ ({𝐶, 𝐴} = {𝐵, 𝐶} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
7		simpl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉)
8		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
9	7, 8	preq1b 4806	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
10	9	biimpcd 252	. . . . . . . . . . . . 13 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵))
11	6, 10	sylbi 220	. . . . . . . . . . . 12 ⊢ ({𝐶, 𝐴} = {𝐵, 𝐶} → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵))
12	11	eqcoms 2773	. . . . . . . . . . 11 ⊢ ({𝐵, 𝐶} = {𝐶, 𝐴} → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵))
13	4, 12	biimtrdi 256	. . . . . . . . . 10 ⊢ ({𝐶, 𝐴} = (𝐼‘𝐾) → ({𝐵, 𝐶} = (𝐼‘𝐾) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵)))
14	13	adantl 486	. . . . . . . . 9 ⊢ ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼‘𝐾)) → ({𝐵, 𝐶} = (𝐼‘𝐾) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵)))
15	14	com12 33	. . . . . . . 8 ⊢ ({𝐵, 𝐶} = (𝐼‘𝐾) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼‘𝐾)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵)))
16	2, 15	biimtrdi 256	. . . . . . 7 ⊢ (𝐽 = 𝐾 → ({𝐵, 𝐶} = (𝐼‘𝐽) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼‘𝐾)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵))))
17	16	adantld 495	. . . . . 6 ⊢ (𝐽 = 𝐾 → ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼‘𝐽)) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼‘𝐾)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵))))
18	17	com14 97	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼‘𝐽)) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼‘𝐾)) → (𝐽 = 𝐾 → 𝐴 = 𝐵))))
19	18	imp32 423	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼‘𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼‘𝐾)))) → (𝐽 = 𝐾 → 𝐴 = 𝐵))
20	19	necon3d 2981	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼‘𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼‘𝐾)))) → (𝐴 ≠ 𝐵 → 𝐽 ≠ 𝐾))
21	20	impancom 456	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼‘𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼‘𝐾))) → 𝐽 ≠ 𝐾))
22	21	imp 411	1 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼‘𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼‘𝐾)))) → 𝐽 ≠ 𝐾)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  {cpr 4587  dom cdm 5651  ‘cfv 6525  Vtxcvtx 29251  iEdgciedg 29252  Edgcedg 29302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-iota 6481  df-fv 6533

This theorem is referenced by:  uhgr3cyclex  30438