Theorem frgrncvvdeqlem7 30622

Description: Lemma 7 for frgrncvvdeq 30626. 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 30620	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
12		fvex 6894	. . . . 5 ⊢ (𝐴‘𝑥) ∈ V
13	12	snid 4627	. . . 4 ⊢ (𝐴‘𝑥) ∈ {(𝐴‘𝑥)}
14		eleq2 2850	. . . . . 6 ⊢ ({(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝐴‘𝑥) ∈ {(𝐴‘𝑥)} ↔ (𝐴‘𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁)))
15	14	biimpa 481	. . . . 5 ⊢ (({(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) ∧ (𝐴‘𝑥) ∈ {(𝐴‘𝑥)}) → (𝐴‘𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
16		elin 3920	. . . . . 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 30616	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∉ 𝑁)
18		df-nel 3063	. . . . . . . . . 10 ⊢ (𝑋 ∉ 𝑁 ↔ ¬ 𝑋 ∈ 𝑁)
19		nelelne 3057	. . . . . . . . . 10 ⊢ (¬ 𝑋 ∈ 𝑁 → ((𝐴‘𝑥) ∈ 𝑁 → (𝐴‘𝑥) ≠ 𝑋))
20	18, 19	sylbi 220	. . . . . . . . 9 ⊢ (𝑋 ∉ 𝑁 → ((𝐴‘𝑥) ∈ 𝑁 → (𝐴‘𝑥) ≠ 𝑋))
21	17, 20	syl 18	. . . . . . . 8 ⊢ (𝜑 → ((𝐴‘𝑥) ∈ 𝑁 → (𝐴‘𝑥) ≠ 𝑋))
22	21	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐴‘𝑥) ∈ 𝑁 → (𝐴‘𝑥) ≠ 𝑋))
23	22	com12 33	. . . . . 6 ⊢ ((𝐴‘𝑥) ∈ 𝑁 → ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋))
24	16, 23	simplbiim 513	. . . . 5 ⊢ ((𝐴‘𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋))
25	15, 24	syl 18	. . . 4 ⊢ (({(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) ∧ (𝐴‘𝑥) ∈ {(𝐴‘𝑥)}) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋))
26	13, 25	mpan2 703	. . 3 ⊢ ({(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋))
27	11, 26	mpcom 39	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋)
28	27	ralrimiva 3155	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2141   ≠ wne 2956   ∉ wnel 3062  ∀wral 3077   ∩ cin 3903  {csn 4588  {cpr 4590   ↦ cmpt 5191  ‘cfv 6536  ℩crio 7366  (class class class)co 7410  Vtxcvtx 29312  Edgcedg 29363   NeighbVtx cnbgr 29648   FriendGraph cfrgr 30575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  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-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-oadd 8456  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-dju 9886  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-n0 12504  df-xnn0 12577  df-z 12591  df-uz 12862  df-fz 13535  df-hash 14366  df-edg 29364  df-upgr 29398  df-umgr 29399  df-usgr 29467  df-nbgr 29649  df-frgr 30576

This theorem is referenced by: (None)