Theorem grlimgrtrilem1 48766

Description: Lemma 3 for grlimgrtri 48768. (Contributed by AV, 24-Aug-2025.) (Proof shortened by AV, 27-Dec-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		simpl 487	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → 𝐺 ∈ UHGraph)
2		simp1 1154	. . . 4 ⊢ (({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼) → {𝑎, 𝑏} ∈ 𝐼)
3	2	adantl 486	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑎, 𝑏} ∈ 𝐼)
4		vex 3459	. . . . 5 ⊢ 𝑎 ∈ V
5	4	prid1 4728	. . . 4 ⊢ 𝑎 ∈ {𝑎, 𝑏}
6	5	a1i 11	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → 𝑎 ∈ {𝑎, 𝑏})
7		grlimgrtrilem1.n	. . . 4 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑎)
8		grlimgrtrilem1.i	. . . 4 ⊢ 𝐼 = (Edg‘𝐺)
9		grlimgrtrilem1.k	. . . 4 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
10	7, 8, 9	clnbgrvtxedg 48759	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ {𝑎, 𝑏} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑏}) → {𝑎, 𝑏} ∈ 𝐾)
11	1, 3, 6, 10	syl3anc 1398	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑎, 𝑏} ∈ 𝐾)
12		simp2 1155	. . . 4 ⊢ (({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼) → {𝑎, 𝑐} ∈ 𝐼)
13	12	adantl 486	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑎, 𝑐} ∈ 𝐼)
14	4	prid1 4728	. . . 4 ⊢ 𝑎 ∈ {𝑎, 𝑐}
15	14	a1i 11	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → 𝑎 ∈ {𝑎, 𝑐})
16	7, 8, 9	clnbgrvtxedg 48759	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑐}) → {𝑎, 𝑐} ∈ 𝐾)
17	1, 13, 15, 16	syl3anc 1398	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑎, 𝑐} ∈ 𝐾)
18		simpr3 1215	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑏, 𝑐} ∈ 𝐼)
19	5	a1i 11	. . . . . 6 ⊢ (({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼) → 𝑎 ∈ {𝑎, 𝑏})
20		vex 3459	. . . . . . . 8 ⊢ 𝑏 ∈ V
21	20	prid2 4729	. . . . . . 7 ⊢ 𝑏 ∈ {𝑎, 𝑏}
22	21	a1i 11	. . . . . 6 ⊢ (({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼) → 𝑏 ∈ {𝑎, 𝑏})
23	2, 19, 22	3jca 1146	. . . . 5 ⊢ (({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼) → ({𝑎, 𝑏} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑏} ∧ 𝑏 ∈ {𝑎, 𝑏}))
24	8, 7	clnbgredg 48605	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑏} ∧ 𝑏 ∈ {𝑎, 𝑏})) → 𝑏 ∈ 𝑁)
25	23, 24	sylan2 604	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → 𝑏 ∈ 𝑁)
26	14	a1i 11	. . . . . 6 ⊢ (({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼) → 𝑎 ∈ {𝑎, 𝑐})
27		vex 3459	. . . . . . . 8 ⊢ 𝑐 ∈ V
28	27	prid2 4729	. . . . . . 7 ⊢ 𝑐 ∈ {𝑎, 𝑐}
29	28	a1i 11	. . . . . 6 ⊢ (({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼) → 𝑐 ∈ {𝑎, 𝑐})
30	12, 26, 29	3jca 1146	. . . . 5 ⊢ (({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼) → ({𝑎, 𝑐} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑐} ∧ 𝑐 ∈ {𝑎, 𝑐}))
31	8, 7	clnbgredg 48605	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑐} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑐} ∧ 𝑐 ∈ {𝑎, 𝑐})) → 𝑐 ∈ 𝑁)
32	30, 31	sylan2 604	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → 𝑐 ∈ 𝑁)
33	25, 32	prssd 4788	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑏, 𝑐} ⊆ 𝑁)
34		sseq1 3962	. . . 4 ⊢ (𝑥 = {𝑏, 𝑐} → (𝑥 ⊆ 𝑁 ↔ {𝑏, 𝑐} ⊆ 𝑁))
35	34, 9	elrab2 3654	. . 3 ⊢ ({𝑏, 𝑐} ∈ 𝐾 ↔ ({𝑏, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ⊆ 𝑁))
36	18, 33, 35	sylanbrc 594	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑏, 𝑐} ∈ 𝐾)
37	11, 17, 36	3jca 1146	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 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  {crab 3416   ⊆ wss 3905  {cpr 4591  ‘cfv 6536  (class class class)co 7410  Vtxcvtx 29346  Edgcedg 29397  UHGraphcuhgr 29406   ClNeighbVtx cclnbgr 48583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-edg 29398  df-uhgr 29408  df-clnbgr 48584

This theorem is referenced by:  grlimgrtri  48768