Theorem frgrncvvdeqlem7 30337

Description: Lemma 7 for frgrncvvdeq 30341. 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 30335	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
12		fvex 6933	. . . . 5 ⊢ (𝐴‘𝑥) ∈ V
13	12	snid 4684	. . . 4 ⊢ (𝐴‘𝑥) ∈ {(𝐴‘𝑥)}
14		eleq2 2833	. . . . . 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 3992	. . . . . 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 30331	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∉ 𝑁)
18		df-nel 3053	. . . . . . . . . 10 ⊢ (𝑋 ∉ 𝑁 ↔ ¬ 𝑋 ∈ 𝑁)
19		nelelne 3047	. . . . . . . . . 10 ⊢ (¬ 𝑋 ∈ 𝑁 → ((𝐴‘𝑥) ∈ 𝑁 → (𝐴‘𝑥) ≠ 𝑋))
20	18, 19	sylbi 217	. . . . . . . . 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 690	. . 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 3152	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   ∉ wnel 3052  ∀wral 3067   ∩ cin 3975  {csn 4648  {cpr 4650   ↦ cmpt 5249  ‘cfv 6573  ℩crio 7403  (class class class)co 7448  Vtxcvtx 29031  Edgcedg 29082   NeighbVtx cnbgr 29367   FriendGraph cfrgr 30290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-fz 13568  df-hash 14380  df-edg 29083  df-upgr 29117  df-umgr 29118  df-usgr 29186  df-nbgr 29368  df-frgr 30291

This theorem is referenced by: (None)