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Theorem frgrancvvdeqlem2 26558
 Description: Lemma 2 for frgrancvvdeq 26569. (Contributed by Alexander van der Vekens, 23-Dec-2017.)
Hypotheses
Ref Expression
frgrancvvdeq.nx 𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)
frgrancvvdeq.ny 𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)
frgrancvvdeq.x (𝜑𝑋𝑉)
frgrancvvdeq.y (𝜑𝑌𝑉)
frgrancvvdeq.ne (𝜑𝑋𝑌)
frgrancvvdeq.xy (𝜑𝑌𝐷)
frgrancvvdeq.f (𝜑𝑉 FriendGrph 𝐸)
frgrancvvdeq.a 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))
Assertion
Ref Expression
frgrancvvdeqlem2 (𝜑𝑋𝑁)
Distinct variable groups:   𝑦,𝐷   𝑥,𝑦,𝑉   𝑥,𝐸,𝑦   𝑦,𝑌   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥,𝑦)   𝐷(𝑥)   𝑁(𝑥,𝑦)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem frgrancvvdeqlem2
StepHypRef Expression
1 frgrancvvdeq.f . . . . . 6 (𝜑𝑉 FriendGrph 𝐸)
2 frisusgra 26519 . . . . . 6 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
3 nbgraeledg 25959 . . . . . 6 (𝑉 USGrph 𝐸 → (𝑋 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑌) ↔ {𝑋, 𝑌} ∈ ran 𝐸))
41, 2, 33syl 18 . . . . 5 (𝜑 → (𝑋 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑌) ↔ {𝑋, 𝑌} ∈ ran 𝐸))
5 frgrancvvdeq.xy . . . . . 6 (𝜑𝑌𝐷)
6 df-nel 2783 . . . . . . . . 9 (𝑌𝐷 ↔ ¬ 𝑌𝐷)
7 frgrancvvdeq.nx . . . . . . . . . 10 𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)
87eleq2i 2680 . . . . . . . . 9 (𝑌𝐷𝑌 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))
96, 8xchbinx 323 . . . . . . . 8 (𝑌𝐷 ↔ ¬ 𝑌 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))
10 nbgraeledg 25959 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → (𝑌 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) ↔ {𝑌, 𝑋} ∈ ran 𝐸))
111, 2, 103syl 18 . . . . . . . . 9 (𝜑 → (𝑌 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) ↔ {𝑌, 𝑋} ∈ ran 𝐸))
1211notbid 307 . . . . . . . 8 (𝜑 → (¬ 𝑌 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) ↔ ¬ {𝑌, 𝑋} ∈ ran 𝐸))
139, 12syl5bb 271 . . . . . . 7 (𝜑 → (𝑌𝐷 ↔ ¬ {𝑌, 𝑋} ∈ ran 𝐸))
14 prcom 4211 . . . . . . . . 9 {𝑌, 𝑋} = {𝑋, 𝑌}
1514eleq1i 2679 . . . . . . . 8 ({𝑌, 𝑋} ∈ ran 𝐸 ↔ {𝑋, 𝑌} ∈ ran 𝐸)
16 pm2.21 119 . . . . . . . 8 (¬ {𝑋, 𝑌} ∈ ran 𝐸 → ({𝑋, 𝑌} ∈ ran 𝐸𝑋 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑌)))
1715, 16sylnbi 319 . . . . . . 7 (¬ {𝑌, 𝑋} ∈ ran 𝐸 → ({𝑋, 𝑌} ∈ ran 𝐸𝑋 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑌)))
1813, 17syl6bi 242 . . . . . 6 (𝜑 → (𝑌𝐷 → ({𝑋, 𝑌} ∈ ran 𝐸𝑋 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑌))))
195, 18mpd 15 . . . . 5 (𝜑 → ({𝑋, 𝑌} ∈ ran 𝐸𝑋 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑌)))
204, 19sylbid 229 . . . 4 (𝜑 → (𝑋 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑌) → 𝑋 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑌)))
2120com12 32 . . 3 (𝑋 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑌) → (𝜑𝑋 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑌)))
22 df-nel 2783 . . . 4 (𝑋 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑌) ↔ ¬ 𝑋 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑌))
23 ax-1 6 . . . 4 (𝑋 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑌) → (𝜑𝑋 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑌)))
2422, 23sylbir 224 . . 3 𝑋 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑌) → (𝜑𝑋 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑌)))
2521, 24pm2.61i 175 . 2 (𝜑𝑋 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑌))
26 eqidd 2611 . . 3 (𝜑𝑋 = 𝑋)
27 frgrancvvdeq.ny . . . 4 𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)
2827a1i 11 . . 3 (𝜑𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌))
2926, 28neleq12d 2887 . 2 (𝜑 → (𝑋𝑁𝑋 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑌)))
3025, 29mpbird 246 1 (𝜑𝑋𝑁)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  {cpr 4127  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ℩crio 6510  (class class class)co 6549   USGrph cusg 25859   Neighbors cnbgra 25946   FriendGrph cfrgra 26515 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-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-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-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-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  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-usgra 25862  df-nbgra 25949  df-frgra 26516 This theorem is referenced by:  frgrancvvdeqlemA  26564  frgrancvvdeqlemB  26565  frgrancvvdeqlemC  26566
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