Theorem grlimgrtrilem1 47808

Description: Lemma 3 for grlimgrtri 47810. (Contributed by AV, 24-Aug-2025.)

Hypotheses

Ref	Expression
grlimgrtrilem1.v	⊢ 𝑉 = (Vtx‘𝐺)
grlimgrtrilem1.n	⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑎)
grlimgrtrilem1.i	⊢ 𝐼 = (Edg‘𝐺)
grlimgrtrilem1.k	⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}

Assertion

Ref	Expression
grlimgrtrilem1	⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → ({𝑎, 𝑏} ∈ 𝐾 ∧ {𝑎, 𝑐} ∈ 𝐾 ∧ {𝑏, 𝑐} ∈ 𝐾))

Distinct variable groups:   𝑥,𝐼   𝑥,𝑁   𝑥,𝑎   𝑥,𝑏   𝑥,𝑐

Allowed substitution hints:   𝐺(𝑥,𝑎,𝑏,𝑐)   𝐼(𝑎,𝑏,𝑐)   𝐾(𝑥,𝑎,𝑏,𝑐)   𝑁(𝑎,𝑏,𝑐)   𝑉(𝑥,𝑎,𝑏,𝑐)

Proof of Theorem grlimgrtrilem1

Step	Hyp	Ref	Expression
1		simpr1 1194	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑎, 𝑏} ∈ 𝐼)
2		simpl 482	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → 𝐺 ∈ UHGraph)
3		vex 3492	. . . . . 6 ⊢ 𝑎 ∈ V
4	3	prid1 4787	. . . . 5 ⊢ 𝑎 ∈ {𝑎, 𝑏}
5	4	a1i 11	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → 𝑎 ∈ {𝑎, 𝑏})
6		grlimgrtrilem1.i	. . . . 5 ⊢ 𝐼 = (Edg‘𝐺)
7		grlimgrtrilem1.n	. . . . 5 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑎)
8	6, 7	clnbgrssedg 47703	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ {𝑎, 𝑏} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑏}) → {𝑎, 𝑏} ⊆ 𝑁)
9	2, 1, 5, 8	syl3anc 1371	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑎, 𝑏} ⊆ 𝑁)
10	1, 9	jca 511	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑏} ⊆ 𝑁))
11		simpr2 1195	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑎, 𝑐} ∈ 𝐼)
12	3	prid1 4787	. . . . 5 ⊢ 𝑎 ∈ {𝑎, 𝑐}
13	12	a1i 11	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → 𝑎 ∈ {𝑎, 𝑐})
14	6, 7	clnbgrssedg 47703	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑐}) → {𝑎, 𝑐} ⊆ 𝑁)
15	2, 11, 13, 14	syl3anc 1371	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑎, 𝑐} ⊆ 𝑁)
16	11, 15	jca 511	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → ({𝑎, 𝑐} ∈ 𝐼 ∧ {𝑎, 𝑐} ⊆ 𝑁))
17		simpr3 1196	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑏, 𝑐} ∈ 𝐼)
18		id 22	. . . . . . 7 ⊢ ({𝑎, 𝑏} ∈ 𝐼 → {𝑎, 𝑏} ∈ 𝐼)
19	4	a1i 11	. . . . . . 7 ⊢ ({𝑎, 𝑏} ∈ 𝐼 → 𝑎 ∈ {𝑎, 𝑏})
20		vex 3492	. . . . . . . . 9 ⊢ 𝑏 ∈ V
21	20	prid2 4788	. . . . . . . 8 ⊢ 𝑏 ∈ {𝑎, 𝑏}
22	21	a1i 11	. . . . . . 7 ⊢ ({𝑎, 𝑏} ∈ 𝐼 → 𝑏 ∈ {𝑎, 𝑏})
23	18, 19, 22	3jca 1128	. . . . . 6 ⊢ ({𝑎, 𝑏} ∈ 𝐼 → ({𝑎, 𝑏} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑏} ∧ 𝑏 ∈ {𝑎, 𝑏}))
24	23	3ad2ant1 1133	. . . . 5 ⊢ (({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼) → ({𝑎, 𝑏} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑏} ∧ 𝑏 ∈ {𝑎, 𝑏}))
25	6, 7	clnbgredg 47702	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑏} ∧ 𝑏 ∈ {𝑎, 𝑏})) → 𝑏 ∈ 𝑁)
26	24, 25	sylan2 592	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → 𝑏 ∈ 𝑁)
27		id 22	. . . . . . 7 ⊢ ({𝑎, 𝑐} ∈ 𝐼 → {𝑎, 𝑐} ∈ 𝐼)
28	12	a1i 11	. . . . . . 7 ⊢ ({𝑎, 𝑐} ∈ 𝐼 → 𝑎 ∈ {𝑎, 𝑐})
29		vex 3492	. . . . . . . . 9 ⊢ 𝑐 ∈ V
30	29	prid2 4788	. . . . . . . 8 ⊢ 𝑐 ∈ {𝑎, 𝑐}
31	30	a1i 11	. . . . . . 7 ⊢ ({𝑎, 𝑐} ∈ 𝐼 → 𝑐 ∈ {𝑎, 𝑐})
32	27, 28, 31	3jca 1128	. . . . . 6 ⊢ ({𝑎, 𝑐} ∈ 𝐼 → ({𝑎, 𝑐} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑐} ∧ 𝑐 ∈ {𝑎, 𝑐}))
33	32	3ad2ant2 1134	. . . . 5 ⊢ (({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼) → ({𝑎, 𝑐} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑐} ∧ 𝑐 ∈ {𝑎, 𝑐}))
34	6, 7	clnbgredg 47702	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑐} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑐} ∧ 𝑐 ∈ {𝑎, 𝑐})) → 𝑐 ∈ 𝑁)
35	33, 34	sylan2 592	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → 𝑐 ∈ 𝑁)
36	26, 35	prssd 4847	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑏, 𝑐} ⊆ 𝑁)
37	17, 36	jca 511	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → ({𝑏, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ⊆ 𝑁))
38		grlimgrtrilem1.k	. . . . 5 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
39	38	eleq2i 2836	. . . 4 ⊢ ({𝑎, 𝑏} ∈ 𝐾 ↔ {𝑎, 𝑏} ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁})
40		sseq1 4034	. . . . 5 ⊢ (𝑥 = {𝑎, 𝑏} → (𝑥 ⊆ 𝑁 ↔ {𝑎, 𝑏} ⊆ 𝑁))
41	40	elrab 3708	. . . 4 ⊢ ({𝑎, 𝑏} ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁} ↔ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑏} ⊆ 𝑁))
42	39, 41	bitri 275	. . 3 ⊢ ({𝑎, 𝑏} ∈ 𝐾 ↔ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑏} ⊆ 𝑁))
43	38	eleq2i 2836	. . . 4 ⊢ ({𝑎, 𝑐} ∈ 𝐾 ↔ {𝑎, 𝑐} ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁})
44		sseq1 4034	. . . . 5 ⊢ (𝑥 = {𝑎, 𝑐} → (𝑥 ⊆ 𝑁 ↔ {𝑎, 𝑐} ⊆ 𝑁))
45	44	elrab 3708	. . . 4 ⊢ ({𝑎, 𝑐} ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁} ↔ ({𝑎, 𝑐} ∈ 𝐼 ∧ {𝑎, 𝑐} ⊆ 𝑁))
46	43, 45	bitri 275	. . 3 ⊢ ({𝑎, 𝑐} ∈ 𝐾 ↔ ({𝑎, 𝑐} ∈ 𝐼 ∧ {𝑎, 𝑐} ⊆ 𝑁))
47	38	eleq2i 2836	. . . 4 ⊢ ({𝑏, 𝑐} ∈ 𝐾 ↔ {𝑏, 𝑐} ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁})
48		sseq1 4034	. . . . 5 ⊢ (𝑥 = {𝑏, 𝑐} → (𝑥 ⊆ 𝑁 ↔ {𝑏, 𝑐} ⊆ 𝑁))
49	48	elrab 3708	. . . 4 ⊢ ({𝑏, 𝑐} ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁} ↔ ({𝑏, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ⊆ 𝑁))
50	47, 49	bitri 275	. . 3 ⊢ ({𝑏, 𝑐} ∈ 𝐾 ↔ ({𝑏, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ⊆ 𝑁))
51	42, 46, 50	3anbi123i 1155	. 2 ⊢ (({𝑎, 𝑏} ∈ 𝐾 ∧ {𝑎, 𝑐} ∈ 𝐾 ∧ {𝑏, 𝑐} ∈ 𝐾) ↔ (({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑏} ⊆ 𝑁) ∧ ({𝑎, 𝑐} ∈ 𝐼 ∧ {𝑎, 𝑐} ⊆ 𝑁) ∧ ({𝑏, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ⊆ 𝑁)))
52	10, 16, 37, 51	syl3anbrc 1343	1 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → ({𝑎, 𝑏} ∈ 𝐾 ∧ {𝑎, 𝑐} ∈ 𝐾 ∧ {𝑏, 𝑐} ∈ 𝐾))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  {crab 3443   ⊆ wss 3976  {cpr 4650  ‘cfv 6568  (class class class)co 7443  Vtxcvtx 29023  Edgcedg 29074  UHGraphcuhgr 29083   ClNeighbVtx cclnbgr 47682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7764

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5701  df-rel 5702  df-cnv 5703  df-co 5704  df-dm 5705  df-rn 5706  df-res 5707  df-ima 5708  df-iota 6520  df-fun 6570  df-fn 6571  df-f 6572  df-fv 6576  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8024  df-2nd 8025  df-edg 29075  df-uhgr 29085  df-clnbgr 47683

This theorem is referenced by:  grlimgrtri  47810