Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frgrncvvdeqlemA Structured version   Visualization version   GIF version

Theorem frgrncvvdeqlemA 41476
 Description: Lemma A for frgrncvvdeq 41480. 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
frgrncvvdeqlemA (𝜑 → ∀𝑥𝐷 (𝐴𝑥) ≠ 𝑋)
Distinct variable groups:   𝑦,𝐷   𝑦,𝐺   𝑦,𝑉   𝑦,𝑌   𝜑,𝑦,𝑥   𝑦,𝐸   𝑦,𝑁   𝑥,𝐷   𝑥,𝑁   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐸(𝑥)   𝐺(𝑥)   𝑉(𝑥)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem frgrncvvdeqlemA
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, 10frgrncvvdeqlem6 41474 . . 3 ((𝜑𝑥𝐷) → {(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
12 fvex 6113 . . . . 5 (𝐴𝑥) ∈ V
1312snid 4155 . . . 4 (𝐴𝑥) ∈ {(𝐴𝑥)}
14 eleq2 2677 . . . . . 6 ({(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝐴𝑥) ∈ {(𝐴𝑥)} ↔ (𝐴𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁)))
1514biimpa 500 . . . . 5 (({(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) ∧ (𝐴𝑥) ∈ {(𝐴𝑥)}) → (𝐴𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
16 elin 3758 . . . . . 6 ((𝐴𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) ↔ ((𝐴𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴𝑥) ∈ 𝑁))
171, 2, 3, 4, 5, 6, 7, 8, 9, 10frgrncvvdeqlem2 41470 . . . . . . . . 9 (𝜑𝑋𝑁)
18 df-nel 2783 . . . . . . . . . 10 (𝑋𝑁 ↔ ¬ 𝑋𝑁)
19 nelelne 2880 . . . . . . . . . 10 𝑋𝑁 → ((𝐴𝑥) ∈ 𝑁 → (𝐴𝑥) ≠ 𝑋))
2018, 19sylbi 206 . . . . . . . . 9 (𝑋𝑁 → ((𝐴𝑥) ∈ 𝑁 → (𝐴𝑥) ≠ 𝑋))
2117, 20syl 17 . . . . . . . 8 (𝜑 → ((𝐴𝑥) ∈ 𝑁 → (𝐴𝑥) ≠ 𝑋))
2221adantr 480 . . . . . . 7 ((𝜑𝑥𝐷) → ((𝐴𝑥) ∈ 𝑁 → (𝐴𝑥) ≠ 𝑋))
2322com12 32 . . . . . 6 ((𝐴𝑥) ∈ 𝑁 → ((𝜑𝑥𝐷) → (𝐴𝑥) ≠ 𝑋))
2416, 23simplbiim 657 . . . . 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 703 . . 3 ({(𝐴𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝜑𝑥𝐷) → (𝐴𝑥) ≠ 𝑋))
2711, 26mpcom 37 . 2 ((𝜑𝑥𝐷) → (𝐴𝑥) ≠ 𝑋)
2827ralrimiva 2949 1 (𝜑 → ∀𝑥𝐷 (𝐴𝑥) ≠ 𝑋)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896   ∩ cin 3539  {csn 4125  {cpr 4127   ↦ cmpt 4643  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Vtxcvtx 25673  Edgcedga 25792   NeighbVtx cnbgr 40550   FriendGraph cfrgr 41428 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-upgr 25749  df-umgr 25750  df-edga 25793  df-usgr 40381  df-nbgr 40554  df-frgr 41429 This theorem is referenced by: (None)
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