Theorem frgrncvvdeqlem7 28570

Description: Lemma 7 for frgrncvvdeq 28574. This corresponds to statement 1 in [Huneke] p. 1: "This common neighbor cannot be x, as x and y are not adjacent.". This is only an observation, which is not required to proof the friendship theorem. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 10-May-2021.)

Hypotheses

Ref	Expression
frgrncvvdeq.v1	⊢ 𝑉 = (Vtx‘𝐺)
frgrncvvdeq.e	⊢ 𝐸 = (Edg‘𝐺)
frgrncvvdeq.nx	⊢ 𝐷 = (𝐺 NeighbVtx 𝑋)
frgrncvvdeq.ny	⊢ 𝑁 = (𝐺 NeighbVtx 𝑌)
frgrncvvdeq.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
frgrncvvdeq.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
frgrncvvdeq.ne	⊢ (𝜑 → 𝑋 ≠ 𝑌)
frgrncvvdeq.xy	⊢ (𝜑 → 𝑌 ∉ 𝐷)
frgrncvvdeq.f	⊢ (𝜑 → 𝐺 ∈ FriendGraph )
frgrncvvdeq.a	⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))

Assertion

Ref	Expression
frgrncvvdeqlem7	⊢ (𝜑 → ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ 𝑋)

Distinct variable groups:   𝑦,𝐸   𝑦,𝐺   𝑦,𝑉   𝑦,𝑌   𝑥,𝑦,𝑁   𝑥,𝐷   𝑥,𝑁   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑥,𝑦)   𝐷(𝑦)   𝐸(𝑥)   𝐺(𝑥)   𝑉(𝑥)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem frgrncvvdeqlem7

Step	Hyp	Ref	Expression
1		frgrncvvdeq.v1	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		frgrncvvdeq.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
3		frgrncvvdeq.nx	. . . 4 ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋)
4		frgrncvvdeq.ny	. . . 4 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌)
5		frgrncvvdeq.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
6		frgrncvvdeq.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
7		frgrncvvdeq.ne	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
8		frgrncvvdeq.xy	. . . 4 ⊢ (𝜑 → 𝑌 ∉ 𝐷)
9		frgrncvvdeq.f	. . . 4 ⊢ (𝜑 → 𝐺 ∈ FriendGraph )
10		frgrncvvdeq.a	. . . 4 ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	frgrncvvdeqlem5 28568	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
12		fvex 6769	. . . . 5 ⊢ (𝐴‘𝑥) ∈ V
13	12	snid 4594	. . . 4 ⊢ (𝐴‘𝑥) ∈ {(𝐴‘𝑥)}
14		eleq2 2827	. . . . . 6 ⊢ ({(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝐴‘𝑥) ∈ {(𝐴‘𝑥)} ↔ (𝐴‘𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁)))
15	14	biimpa 476	. . . . 5 ⊢ (({(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) ∧ (𝐴‘𝑥) ∈ {(𝐴‘𝑥)}) → (𝐴‘𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
16		elin 3899	. . . . . 6 ⊢ ((𝐴‘𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) ↔ ((𝐴‘𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴‘𝑥) ∈ 𝑁))
17	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	frgrncvvdeqlem1 28564	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∉ 𝑁)
18		df-nel 3049	. . . . . . . . . 10 ⊢ (𝑋 ∉ 𝑁 ↔ ¬ 𝑋 ∈ 𝑁)
19		nelelne 3042	. . . . . . . . . 10 ⊢ (¬ 𝑋 ∈ 𝑁 → ((𝐴‘𝑥) ∈ 𝑁 → (𝐴‘𝑥) ≠ 𝑋))
20	18, 19	sylbi 216	. . . . . . . . 9 ⊢ (𝑋 ∉ 𝑁 → ((𝐴‘𝑥) ∈ 𝑁 → (𝐴‘𝑥) ≠ 𝑋))
21	17, 20	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝐴‘𝑥) ∈ 𝑁 → (𝐴‘𝑥) ≠ 𝑋))
22	21	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐴‘𝑥) ∈ 𝑁 → (𝐴‘𝑥) ≠ 𝑋))
23	22	com12 32	. . . . . 6 ⊢ ((𝐴‘𝑥) ∈ 𝑁 → ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋))
24	16, 23	simplbiim 504	. . . . 5 ⊢ ((𝐴‘𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋))
25	15, 24	syl 17	. . . 4 ⊢ (({(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) ∧ (𝐴‘𝑥) ∈ {(𝐴‘𝑥)}) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋))
26	13, 25	mpan2 687	. . 3 ⊢ ({(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋))
27	11, 26	mpcom 38	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋)
28	27	ralrimiva 3107	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942   ∉ wnel 3048  ∀wral 3063   ∩ cin 3882  {csn 4558  {cpr 4560   ↦ cmpt 5153  ‘cfv 6418  ℩crio 7211  (class class class)co 7255  Vtxcvtx 27269  Edgcedg 27320   NeighbVtx cnbgr 27602   FriendGraph cfrgr 28523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-fz 13169  df-hash 13973  df-edg 27321  df-upgr 27355  df-umgr 27356  df-usgr 27424  df-nbgr 27603  df-frgr 28524

This theorem is referenced by: (None)