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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 = {𝑥𝐼𝑥𝑁}
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
StepHypRef Expression
1 simpr1 1195 . . 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 3483 . . . . . 6 𝑎 ∈ V
43prid1 4760 . . . . 5 𝑎 ∈ {𝑎, 𝑏}
54a1i 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)
86, 7clnbgrssedg 47800 . . . 4 ((𝐺 ∈ UHGraph ∧ {𝑎, 𝑏} ∈ 𝐼𝑎 ∈ {𝑎, 𝑏}) → {𝑎, 𝑏} ⊆ 𝑁)
92, 1, 5, 8syl3anc 1373 . . 3 ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑎, 𝑏} ⊆ 𝑁)
101, 9jca 511 . 2 ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑏} ⊆ 𝑁))
11 simpr2 1196 . . 3 ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑎, 𝑐} ∈ 𝐼)
123prid1 4760 . . . . 5 𝑎 ∈ {𝑎, 𝑐}
1312a1i 11 . . . 4 ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → 𝑎 ∈ {𝑎, 𝑐})
146, 7clnbgrssedg 47800 . . . 4 ((𝐺 ∈ UHGraph ∧ {𝑎, 𝑐} ∈ 𝐼𝑎 ∈ {𝑎, 𝑐}) → {𝑎, 𝑐} ⊆ 𝑁)
152, 11, 13, 14syl3anc 1373 . . 3 ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑎, 𝑐} ⊆ 𝑁)
1611, 15jca 511 . 2 ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → ({𝑎, 𝑐} ∈ 𝐼 ∧ {𝑎, 𝑐} ⊆ 𝑁))
17 simpr3 1197 . . 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)
194a1i 11 . . . . . . 7 ({𝑎, 𝑏} ∈ 𝐼𝑎 ∈ {𝑎, 𝑏})
20 vex 3483 . . . . . . . . 9 𝑏 ∈ V
2120prid2 4761 . . . . . . . 8 𝑏 ∈ {𝑎, 𝑏}
2221a1i 11 . . . . . . 7 ({𝑎, 𝑏} ∈ 𝐼𝑏 ∈ {𝑎, 𝑏})
2318, 19, 223jca 1129 . . . . . 6 ({𝑎, 𝑏} ∈ 𝐼 → ({𝑎, 𝑏} ∈ 𝐼𝑎 ∈ {𝑎, 𝑏} ∧ 𝑏 ∈ {𝑎, 𝑏}))
24233ad2ant1 1134 . . . . 5 (({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼) → ({𝑎, 𝑏} ∈ 𝐼𝑎 ∈ {𝑎, 𝑏} ∧ 𝑏 ∈ {𝑎, 𝑏}))
256, 7clnbgredg 47799 . . . . 5 ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼𝑎 ∈ {𝑎, 𝑏} ∧ 𝑏 ∈ {𝑎, 𝑏})) → 𝑏𝑁)
2624, 25sylan2 593 . . . 4 ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → 𝑏𝑁)
27 id 22 . . . . . . 7 ({𝑎, 𝑐} ∈ 𝐼 → {𝑎, 𝑐} ∈ 𝐼)
2812a1i 11 . . . . . . 7 ({𝑎, 𝑐} ∈ 𝐼𝑎 ∈ {𝑎, 𝑐})
29 vex 3483 . . . . . . . . 9 𝑐 ∈ V
3029prid2 4761 . . . . . . . 8 𝑐 ∈ {𝑎, 𝑐}
3130a1i 11 . . . . . . 7 ({𝑎, 𝑐} ∈ 𝐼𝑐 ∈ {𝑎, 𝑐})
3227, 28, 313jca 1129 . . . . . 6 ({𝑎, 𝑐} ∈ 𝐼 → ({𝑎, 𝑐} ∈ 𝐼𝑎 ∈ {𝑎, 𝑐} ∧ 𝑐 ∈ {𝑎, 𝑐}))
33323ad2ant2 1135 . . . . 5 (({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼) → ({𝑎, 𝑐} ∈ 𝐼𝑎 ∈ {𝑎, 𝑐} ∧ 𝑐 ∈ {𝑎, 𝑐}))
346, 7clnbgredg 47799 . . . . 5 ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑐} ∈ 𝐼𝑎 ∈ {𝑎, 𝑐} ∧ 𝑐 ∈ {𝑎, 𝑐})) → 𝑐𝑁)
3533, 34sylan2 593 . . . 4 ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → 𝑐𝑁)
3626, 35prssd 4820 . . 3 ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → {𝑏, 𝑐} ⊆ 𝑁)
3717, 36jca 511 . 2 ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → ({𝑏, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ⊆ 𝑁))
38 grlimgrtrilem1.k . . . . 5 𝐾 = {𝑥𝐼𝑥𝑁}
3938eleq2i 2832 . . . 4 ({𝑎, 𝑏} ∈ 𝐾 ↔ {𝑎, 𝑏} ∈ {𝑥𝐼𝑥𝑁})
40 sseq1 4008 . . . . 5 (𝑥 = {𝑎, 𝑏} → (𝑥𝑁 ↔ {𝑎, 𝑏} ⊆ 𝑁))
4140elrab 3691 . . . 4 ({𝑎, 𝑏} ∈ {𝑥𝐼𝑥𝑁} ↔ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑏} ⊆ 𝑁))
4239, 41bitri 275 . . 3 ({𝑎, 𝑏} ∈ 𝐾 ↔ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑏} ⊆ 𝑁))
4338eleq2i 2832 . . . 4 ({𝑎, 𝑐} ∈ 𝐾 ↔ {𝑎, 𝑐} ∈ {𝑥𝐼𝑥𝑁})
44 sseq1 4008 . . . . 5 (𝑥 = {𝑎, 𝑐} → (𝑥𝑁 ↔ {𝑎, 𝑐} ⊆ 𝑁))
4544elrab 3691 . . . 4 ({𝑎, 𝑐} ∈ {𝑥𝐼𝑥𝑁} ↔ ({𝑎, 𝑐} ∈ 𝐼 ∧ {𝑎, 𝑐} ⊆ 𝑁))
4643, 45bitri 275 . . 3 ({𝑎, 𝑐} ∈ 𝐾 ↔ ({𝑎, 𝑐} ∈ 𝐼 ∧ {𝑎, 𝑐} ⊆ 𝑁))
4738eleq2i 2832 . . . 4 ({𝑏, 𝑐} ∈ 𝐾 ↔ {𝑏, 𝑐} ∈ {𝑥𝐼𝑥𝑁})
48 sseq1 4008 . . . . 5 (𝑥 = {𝑏, 𝑐} → (𝑥𝑁 ↔ {𝑏, 𝑐} ⊆ 𝑁))
4948elrab 3691 . . . 4 ({𝑏, 𝑐} ∈ {𝑥𝐼𝑥𝑁} ↔ ({𝑏, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ⊆ 𝑁))
5047, 49bitri 275 . . 3 ({𝑏, 𝑐} ∈ 𝐾 ↔ ({𝑏, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ⊆ 𝑁))
5142, 46, 503anbi123i 1156 . 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)))
5210, 16, 37, 51syl3anbrc 1344 1 ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → ({𝑎, 𝑏} ∈ 𝐾 ∧ {𝑎, 𝑐} ∈ 𝐾 ∧ {𝑏, 𝑐} ∈ 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  {crab 3435  wss 3950  {cpr 4626  cfv 6559  (class class class)co 7429  Vtxcvtx 29003  Edgcedg 29054  UHGraphcuhgr 29063   ClNeighbVtx cclnbgr 47778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430  ax-un 7751
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-1st 8010  df-2nd 8011  df-edg 29055  df-uhgr 29065  df-clnbgr 47779
This theorem is referenced by:  grlimgrtri  47936
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