Theorem uhgr3cyclexlem 30151

Description: Lemma for uhgr3cyclex 30152. (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 6817	. . . . . . . . 9 ⊢ (𝐽 = 𝐾 → (𝐼‘𝐽) = (𝐼‘𝐾))
2	1	eqeq2d 2741	. . . . . . . 8 ⊢ (𝐽 = 𝐾 → ({𝐵, 𝐶} = (𝐼‘𝐽) ↔ {𝐵, 𝐶} = (𝐼‘𝐾)))
3		eqeq2 2742	. . . . . . . . . . . 12 ⊢ ((𝐼‘𝐾) = {𝐶, 𝐴} → ({𝐵, 𝐶} = (𝐼‘𝐾) ↔ {𝐵, 𝐶} = {𝐶, 𝐴}))
4	3	eqcoms 2738	. . . . . . . . . . 11 ⊢ ({𝐶, 𝐴} = (𝐼‘𝐾) → ({𝐵, 𝐶} = (𝐼‘𝐾) ↔ {𝐵, 𝐶} = {𝐶, 𝐴}))
5		prcom 4683	. . . . . . . . . . . . . 14 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
6	5	eqeq1i 2735	. . . . . . . . . . . . 13 ⊢ ({𝐶, 𝐴} = {𝐵, 𝐶} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
7		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉)
8		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
9	7, 8	preq1b 4796	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
10	9	biimpcd 249	. . . . . . . . . . . . 13 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵))
11	6, 10	sylbi 217	. . . . . . . . . . . 12 ⊢ ({𝐶, 𝐴} = {𝐵, 𝐶} → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵))
12	11	eqcoms 2738	. . . . . . . . . . 11 ⊢ ({𝐵, 𝐶} = {𝐶, 𝐴} → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵))
13	4, 12	biimtrdi 253	. . . . . . . . . 10 ⊢ ({𝐶, 𝐴} = (𝐼‘𝐾) → ({𝐵, 𝐶} = (𝐼‘𝐾) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵)))
14	13	adantl 481	. . . . . . . . 9 ⊢ ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼‘𝐾)) → ({𝐵, 𝐶} = (𝐼‘𝐾) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵)))
15	14	com12 32	. . . . . . . 8 ⊢ ({𝐵, 𝐶} = (𝐼‘𝐾) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼‘𝐾)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵)))
16	2, 15	biimtrdi 253	. . . . . . 7 ⊢ (𝐽 = 𝐾 → ({𝐵, 𝐶} = (𝐼‘𝐽) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼‘𝐾)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵))))
17	16	adantld 490	. . . . . 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 96	. . . . 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 418	. . . 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 2947	. . 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 451	. 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 406	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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110   ≠ wne 2926  {cpr 4576  dom cdm 5614  ‘cfv 6477  Vtxcvtx 28967  iEdgciedg 28968  Edgcedg 29018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-iota 6433  df-fv 6485

This theorem is referenced by:  uhgr3cyclex  30152