Theorem grlimgrtrilem1 49206

Description: Lemma 3 for grlimgrtri 49208. (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 488	. . 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 487	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑎, 𝑏} ∈ 𝐼)
4		vex 3454	. . . . 5 ⊢ 𝑎 ∈ V
5	4	prid1 4719	. . . 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 49199	. . 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 487	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑎, 𝑐} ∈ 𝐼)
14	4	prid1 4719	. . . 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 49199	. . 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 3454	. . . . . . . 8 ⊢ 𝑏 ∈ V
21	20	prid2 4720	. . . . . . 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 49045	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑏} ∧ 𝑏 ∈ {𝑎, 𝑏})) → 𝑏 ∈ 𝑁)
25	23, 24	sylan2 605	. . . 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 3454	. . . . . . . 8 ⊢ 𝑐 ∈ V
28	27	prid2 4720	. . . . . . 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 49045	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑐} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑐} ∧ 𝑐 ∈ {𝑎, 𝑐})) → 𝑐 ∈ 𝑁)
32	30, 31	sylan2 605	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → 𝑐 ∈ 𝑁)
33	25, 32	prssd 4777	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑏, 𝑐} ⊆ 𝑁)
34		sseq1 3934	. . . 4 ⊢ (𝑥 = {𝑏, 𝑐} → (𝑥 ⊆ 𝑁 ↔ {𝑏, 𝑐} ⊆ 𝑁))
35	34, 9	elrab2 3648	. . 3 ⊢ ({𝑏, 𝑐} ∈ 𝐾 ↔ ({𝑏, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ⊆ 𝑁))
36	18, 33, 35	sylanbrc 595	. 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  {crab 3412   ⊆ wss 3896  {cpr 4586  ‘cfv 6541  (class class class)co 7422  Vtxcvtx 29581  Edgcedg 29632  UHGraphcuhgr 29641   ClNeighbVtx cclnbgr 49023

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7425  df-oprab 7426  df-mpo 7427  df-1st 8003  df-2nd 8004  df-edg 29633  df-uhgr 29643  df-clnbgr 49024

This theorem is used by:  grlimgrtri  49208