Theorem grlimgrtrilem1 48247

Description: Lemma 3 for grlimgrtri 48249. (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 482	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → 𝐺 ∈ UHGraph)
2		simp1 1136	. . . 4 ⊢ (({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼) → {𝑎, 𝑏} ∈ 𝐼)
3	2	adantl 481	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑎, 𝑏} ∈ 𝐼)
4		vex 3444	. . . . 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 48240	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ {𝑎, 𝑏} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑏}) → {𝑎, 𝑏} ∈ 𝐾)
11	1, 3, 6, 10	syl3anc 1373	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑎, 𝑏} ∈ 𝐾)
12		simp2 1137	. . . 4 ⊢ (({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼) → {𝑎, 𝑐} ∈ 𝐼)
13	12	adantl 481	. . 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 48240	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑐}) → {𝑎, 𝑐} ∈ 𝐾)
17	1, 13, 15, 16	syl3anc 1373	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑎, 𝑐} ∈ 𝐾)
18		simpr3 1197	. . 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 3444	. . . . . . . 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 1128	. . . . 5 ⊢ (({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼) → ({𝑎, 𝑏} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑏} ∧ 𝑏 ∈ {𝑎, 𝑏}))
24	8, 7	clnbgredg 48086	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑏} ∧ 𝑏 ∈ {𝑎, 𝑏})) → 𝑏 ∈ 𝑁)
25	23, 24	sylan2 593	. . . 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 3444	. . . . . . . 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 1128	. . . . 5 ⊢ (({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼) → ({𝑎, 𝑐} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑐} ∧ 𝑐 ∈ {𝑎, 𝑐}))
31	8, 7	clnbgredg 48086	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑐} ∈ 𝐼 ∧ 𝑎 ∈ {𝑎, 𝑐} ∧ 𝑐 ∈ {𝑎, 𝑐})) → 𝑐 ∈ 𝑁)
32	30, 31	sylan2 593	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → 𝑐 ∈ 𝑁)
33	25, 32	prssd 4778	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑏, 𝑐} ⊆ 𝑁)
34		sseq1 3959	. . . 4 ⊢ (𝑥 = {𝑏, 𝑐} → (𝑥 ⊆ 𝑁 ↔ {𝑏, 𝑐} ⊆ 𝑁))
35	34, 9	elrab2 3649	. . 3 ⊢ ({𝑏, 𝑐} ∈ 𝐾 ↔ ({𝑏, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ⊆ 𝑁))
36	18, 33, 35	sylanbrc 583	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑏, 𝑐} ∈ 𝐾)
37	11, 17, 36	3jca 1128	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 1541   ∈ wcel 2113  {crab 3399   ⊆ wss 3901  {cpr 4582  ‘cfv 6492  (class class class)co 7358  Vtxcvtx 29069  Edgcedg 29120  UHGraphcuhgr 29129   ClNeighbVtx cclnbgr 48064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-edg 29121  df-uhgr 29131  df-clnbgr 48065

This theorem is referenced by:  grlimgrtri  48249