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Theorem frgrncvvdeqlem7 29823
Description: Lemma 7 for frgrncvvdeq 29827. 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 (𝜑𝑋𝑉)
frgrncvvdeq.y (𝜑𝑌𝑉)
frgrncvvdeq.ne (𝜑𝑋𝑌)
frgrncvvdeq.xy (𝜑𝑌𝐷)
frgrncvvdeq.f (𝜑𝐺 ∈ FriendGraph )
frgrncvvdeq.a 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
Assertion
Ref Expression
frgrncvvdeqlem7 (𝜑 → ∀𝑥𝐷 (𝐴𝑥) ≠ 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
StepHypRef 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 (𝜑𝑋𝑉)
6 frgrncvvdeq.y . . . 4 (𝜑𝑌𝑉)
7 frgrncvvdeq.ne . . . 4 (𝜑𝑋𝑌)
8 frgrncvvdeq.xy . . . 4 (𝜑𝑌𝐷)
9 frgrncvvdeq.f . . . 4 (𝜑𝐺 ∈ FriendGraph )
10 frgrncvvdeq.a . . . 4 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10frgrncvvdeqlem5 29821 . . 3 ((𝜑𝑥𝐷) → {(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
12 fvex 6905 . . . . 5 (𝐴𝑥) ∈ V
1312snid 4665 . . . 4 (𝐴𝑥) ∈ {(𝐴𝑥)}
14 eleq2 2820 . . . . . 6 ({(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝐴𝑥) ∈ {(𝐴𝑥)} ↔ (𝐴𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁)))
1514biimpa 475 . . . . 5 (({(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) ∧ (𝐴𝑥) ∈ {(𝐴𝑥)}) → (𝐴𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
16 elin 3965 . . . . . 6 ((𝐴𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) ↔ ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴𝑥) ∈ 𝑁))
171, 2, 3, 4, 5, 6, 7, 8, 9, 10frgrncvvdeqlem1 29817 . . . . . . . . 9 (𝜑𝑋𝑁)
18 df-nel 3045 . . . . . . . . . 10 (𝑋𝑁 ↔ ¬ 𝑋𝑁)
19 nelelne 3039 . . . . . . . . . 10 𝑋𝑁 → ((𝐴𝑥) ∈ 𝑁 → (𝐴𝑥) ≠ 𝑋))
2018, 19sylbi 216 . . . . . . . . 9 (𝑋𝑁 → ((𝐴𝑥) ∈ 𝑁 → (𝐴𝑥) ≠ 𝑋))
2117, 20syl 17 . . . . . . . 8 (𝜑 → ((𝐴𝑥) ∈ 𝑁 → (𝐴𝑥) ≠ 𝑋))
2221adantr 479 . . . . . . 7 ((𝜑𝑥𝐷) → ((𝐴𝑥) ∈ 𝑁 → (𝐴𝑥) ≠ 𝑋))
2322com12 32 . . . . . 6 ((𝐴𝑥) ∈ 𝑁 → ((𝜑𝑥𝐷) → (𝐴𝑥) ≠ 𝑋))
2416, 23simplbiim 503 . . . . 5 ((𝐴𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝜑𝑥𝐷) → (𝐴𝑥) ≠ 𝑋))
2515, 24syl 17 . . . 4 (({(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) ∧ (𝐴𝑥) ∈ {(𝐴𝑥)}) → ((𝜑𝑥𝐷) → (𝐴𝑥) ≠ 𝑋))
2613, 25mpan2 687 . . 3 ({(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝜑𝑥𝐷) → (𝐴𝑥) ≠ 𝑋))
2711, 26mpcom 38 . 2 ((𝜑𝑥𝐷) → (𝐴𝑥) ≠ 𝑋)
2827ralrimiva 3144 1 (𝜑 → ∀𝑥𝐷 (𝐴𝑥) ≠ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1539  wcel 2104  wne 2938  wnel 3044  wral 3059  cin 3948  {csn 4629  {cpr 4631  cmpt 5232  cfv 6544  crio 7368  (class class class)co 7413  Vtxcvtx 28521  Edgcedg 28572   NeighbVtx cnbgr 28854   FriendGraph cfrgr 29776
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-2o 8471  df-oadd 8474  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-dju 9900  df-card 9938  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11452  df-neg 11453  df-nn 12219  df-2 12281  df-n0 12479  df-xnn0 12551  df-z 12565  df-uz 12829  df-fz 13491  df-hash 14297  df-edg 28573  df-upgr 28607  df-umgr 28608  df-usgr 28676  df-nbgr 28855  df-frgr 29777
This theorem is referenced by: (None)
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