Theorem frgrncvvdeqlem7 30398

Description: Lemma 7 for frgrncvvdeq 30402. 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 30396	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
12		fvex 6857	. . . . 5 ⊢ (𝐴‘𝑥) ∈ V
13	12	snid 4621	. . . 4 ⊢ (𝐴‘𝑥) ∈ {(𝐴‘𝑥)}
14		eleq2 2826	. . . . . 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 3919	. . . . . 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 30392	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∉ 𝑁)
18		df-nel 3038	. . . . . . . . . 10 ⊢ (𝑋 ∉ 𝑁 ↔ ¬ 𝑋 ∈ 𝑁)
19		nelelne 3032	. . . . . . . . . 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 692	. . 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 3130	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   ∉ wnel 3037  ∀wral 3052   ∩ cin 3902  {csn 4582  {cpr 4584   ↦ cmpt 5181  ‘cfv 6502  ℩crio 7326  (class class class)co 7370  Vtxcvtx 29087  Edgcedg 29138   NeighbVtx cnbgr 29423   FriendGraph cfrgr 30351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-2o 8410  df-oadd 8413  df-er 8647  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-dju 9827  df-card 9865  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-n0 12416  df-xnn0 12489  df-z 12503  df-uz 12766  df-fz 13438  df-hash 14268  df-edg 29139  df-upgr 29173  df-umgr 29174  df-usgr 29242  df-nbgr 29424  df-frgr 30352

This theorem is referenced by: (None)