Theorem uhgr3cyclexlem 30276

Description: Lemma for uhgr3cyclex 30277. (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 6834	. . . . . . . . 9 ⊢ (𝐽 = 𝐾 → (𝐼‘𝐽) = (𝐼‘𝐾))
2	1	eqeq2d 2751	. . . . . . . 8 ⊢ (𝐽 = 𝐾 → ({𝐵, 𝐶} = (𝐼‘𝐽) ↔ {𝐵, 𝐶} = (𝐼‘𝐾)))
3		eqeq2 2752	. . . . . . . . . . . 12 ⊢ ((𝐼‘𝐾) = {𝐶, 𝐴} → ({𝐵, 𝐶} = (𝐼‘𝐾) ↔ {𝐵, 𝐶} = {𝐶, 𝐴}))
4	3	eqcoms 2748	. . . . . . . . . . 11 ⊢ ({𝐶, 𝐴} = (𝐼‘𝐾) → ({𝐵, 𝐶} = (𝐼‘𝐾) ↔ {𝐵, 𝐶} = {𝐶, 𝐴}))
5		prcom 4671	. . . . . . . . . . . . . 14 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
6	5	eqeq1i 2745	. . . . . . . . . . . . 13 ⊢ ({𝐶, 𝐴} = {𝐵, 𝐶} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
7		simpl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉)
8		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
9	7, 8	preq1b 4784	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
10	9	biimpcd 250	. . . . . . . . . . . . 13 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵))
11	6, 10	sylbi 218	. . . . . . . . . . . 12 ⊢ ({𝐶, 𝐴} = {𝐵, 𝐶} → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵))
12	11	eqcoms 2748	. . . . . . . . . . 11 ⊢ ({𝐵, 𝐶} = {𝐶, 𝐴} → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵))
13	4, 12	biimtrdi 254	. . . . . . . . . 10 ⊢ ({𝐶, 𝐴} = (𝐼‘𝐾) → ({𝐵, 𝐶} = (𝐼‘𝐾) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵)))
14	13	adantl 482	. . . . . . . . 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 254	. . . . . . 7 ⊢ (𝐽 = 𝐾 → ({𝐵, 𝐶} = (𝐼‘𝐽) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼‘𝐾)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵))))
17	16	adantld 491	. . . . . 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 419	. . . 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 2956	. . 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 452	. 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 407	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 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  {cpr 4564  dom cdm 5625  ‘cfv 6492  Vtxcvtx 29090  iEdgciedg 29091  Edgcedg 29141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500

This theorem is referenced by:  uhgr3cyclex  30277