Theorem uhgr3cyclexlem 27041

Description: Lemma for uhgr3cyclex 27042. (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 6191	. . . . . . . . 9 ⊢ (𝐽 = 𝐾 → (𝐼‘𝐽) = (𝐼‘𝐾))
2	1	eqeq2d 2632	. . . . . . . 8 ⊢ (𝐽 = 𝐾 → ({𝐵, 𝐶} = (𝐼‘𝐽) ↔ {𝐵, 𝐶} = (𝐼‘𝐾)))
3		eqeq2 2633	. . . . . . . . . . . 12 ⊢ ((𝐼‘𝐾) = {𝐶, 𝐴} → ({𝐵, 𝐶} = (𝐼‘𝐾) ↔ {𝐵, 𝐶} = {𝐶, 𝐴}))
4	3	eqcoms 2630	. . . . . . . . . . 11 ⊢ ({𝐶, 𝐴} = (𝐼‘𝐾) → ({𝐵, 𝐶} = (𝐼‘𝐾) ↔ {𝐵, 𝐶} = {𝐶, 𝐴}))
5		prcom 4267	. . . . . . . . . . . . . 14 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
6	5	eqeq1i 2627	. . . . . . . . . . . . 13 ⊢ ({𝐶, 𝐴} = {𝐵, 𝐶} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
7		simpl 473	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉)
8		simpr 477	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
9	7, 8	preq1b 4377	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
10	9	biimpcd 239	. . . . . . . . . . . . 13 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵))
11	6, 10	sylbi 207	. . . . . . . . . . . 12 ⊢ ({𝐶, 𝐴} = {𝐵, 𝐶} → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵))
12	11	eqcoms 2630	. . . . . . . . . . 11 ⊢ ({𝐵, 𝐶} = {𝐶, 𝐴} → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵))
13	4, 12	syl6bi 243	. . . . . . . . . 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	syl6bi 243	. . . . . . 7 ⊢ (𝐽 = 𝐾 → ({𝐵, 𝐶} = (𝐼‘𝐽) → ((𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼‘𝐾)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 = 𝐵))))
17	16	adantld 483	. . . . . 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 449	. . . 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 2815	. . 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 445	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  {cpr 4179  dom cdm 5114  ‘cfv 5888  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  uhgr3cyclex  27042