Theorem grlimgrtrilem1 47934

Description: Lemma 3 for grlimgrtri 47936. (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 3467	. . . . . 6 ⊢ 𝑎 ∈ V
4	3	prid1 4742	. . . . 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 47800	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ {𝑎, 𝑏} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑏}) → {𝑎, 𝑏} ⊆ 𝑁)
9	2, 1, 5, 8	syl3anc 1372	. . 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 4742	. . . . 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 47800	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑐}) → {𝑎, 𝑐} ⊆ 𝑁)
15	2, 11, 13, 14	syl3anc 1372	. . 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 3467	. . . . . . . . 9 ⊢ 𝑏 ∈ V
21	20	prid2 4743	. . . . . . . 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 47799	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑏} ∧ 𝑏 ∈ {𝑎, 𝑏})) → 𝑏 ∈ 𝑁)
26	24, 25	sylan2 593	. . . 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 3467	. . . . . . . . 9 ⊢ 𝑐 ∈ V
30	29	prid2 4743	. . . . . . . 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 47799	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑐} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑐} ∧ 𝑐 ∈ {𝑎, 𝑐})) → 𝑐 ∈ 𝑁)
35	33, 34	sylan2 593	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → 𝑐 ∈ 𝑁)
36	26, 35	prssd 4802	. . 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 2825	. . . 4 ⊢ ({𝑎, 𝑏} ∈ 𝐾 ↔ {𝑎, 𝑏} ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁})
40		sseq1 3989	. . . . 5 ⊢ (𝑥 = {𝑎, 𝑏} → (𝑥 ⊆ 𝑁 ↔ {𝑎, 𝑏} ⊆ 𝑁))
41	40	elrab 3675	. . . 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 2825	. . . 4 ⊢ ({𝑎, 𝑐} ∈ 𝐾 ↔ {𝑎, 𝑐} ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁})
44		sseq1 3989	. . . . 5 ⊢ (𝑥 = {𝑎, 𝑐} → (𝑥 ⊆ 𝑁 ↔ {𝑎, 𝑐} ⊆ 𝑁))
45	44	elrab 3675	. . . 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 2825	. . . 4 ⊢ ({𝑏, 𝑐} ∈ 𝐾 ↔ {𝑏, 𝑐} ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁})
48		sseq1 3989	. . . . 5 ⊢ (𝑥 = {𝑏, 𝑐} → (𝑥 ⊆ 𝑁 ↔ {𝑏, 𝑐} ⊆ 𝑁))
49	48	elrab 3675	. . . 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 1086   = wceq 1539   ∈ wcel 2107  {crab 3419   ⊆ wss 3931  {cpr 4608  ‘cfv 6541  (class class class)co 7413  Vtxcvtx 28942  Edgcedg 28993  UHGraphcuhgr 29002   ClNeighbVtx cclnbgr 47778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7996  df-2nd 7997  df-edg 28994  df-uhgr 29004  df-clnbgr 47779

This theorem is referenced by:  grlimgrtri  47936