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Theorem frgrancvvdeqlemA 26358
Description: Lemma A for frgrancvvdeq 26363. This corresponds to statement 1 in [Huneke] p. 1: "This common neighbor cannot be x, as x and y are not adjacent.". (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
frgrancvvdeqlemA (𝜑 → ∀𝑥𝐷 (𝐴𝑥) ≠ 𝑋)
Distinct variable groups:   𝑦,𝐷,𝑥   𝑥,𝑉,𝑦   𝑥,𝐸,𝑦   𝑦,𝑌   𝜑,𝑦   𝑦,𝑁   𝑥,𝐷   𝑥,𝑁   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem frgrancvvdeqlemA
StepHypRef Expression
1 frgrancvvdeq.nx . . . 4 𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)
2 frgrancvvdeq.ny . . . 4 𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)
3 frgrancvvdeq.x . . . 4 (𝜑𝑋𝑉)
4 frgrancvvdeq.y . . . 4 (𝜑𝑌𝑉)
5 frgrancvvdeq.ne . . . 4 (𝜑𝑋𝑌)
6 frgrancvvdeq.xy . . . 4 (𝜑𝑌𝐷)
7 frgrancvvdeq.f . . . 4 (𝜑𝑉 FriendGrph 𝐸)
8 frgrancvvdeq.a . . . 4 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))
91, 2, 3, 4, 5, 6, 7, 8frgrancvvdeqlem6 26356 . . 3 ((𝜑𝑥𝐷) → {(𝐴𝑥)} = ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁))
10 fvex 6098 . . . . 5 (𝐴𝑥) ∈ V
1110snid 4154 . . . 4 (𝐴𝑥) ∈ {(𝐴𝑥)}
12 eleq2 2676 . . . . . 6 ({(𝐴𝑥)} = ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁) → ((𝐴𝑥) ∈ {(𝐴𝑥)} ↔ (𝐴𝑥) ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁)))
1312biimpa 499 . . . . 5 (({(𝐴𝑥)} = ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁) ∧ (𝐴𝑥) ∈ {(𝐴𝑥)}) → (𝐴𝑥) ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁))
14 elin 3757 . . . . . 6 ((𝐴𝑥) ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁) ↔ ((𝐴𝑥) ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∧ (𝐴𝑥) ∈ 𝑁))
151, 2, 3, 4, 5, 6, 7, 8frgrancvvdeqlem2 26352 . . . . . . . . . 10 (𝜑𝑋𝑁)
16 df-nel 2782 . . . . . . . . . . 11 (𝑋𝑁 ↔ ¬ 𝑋𝑁)
17 eleq1 2675 . . . . . . . . . . . . . 14 ((𝐴𝑥) = 𝑋 → ((𝐴𝑥) ∈ 𝑁𝑋𝑁))
1817biimpcd 237 . . . . . . . . . . . . 13 ((𝐴𝑥) ∈ 𝑁 → ((𝐴𝑥) = 𝑋𝑋𝑁))
1918con3rr3 149 . . . . . . . . . . . 12 𝑋𝑁 → ((𝐴𝑥) ∈ 𝑁 → ¬ (𝐴𝑥) = 𝑋))
20 df-ne 2781 . . . . . . . . . . . 12 ((𝐴𝑥) ≠ 𝑋 ↔ ¬ (𝐴𝑥) = 𝑋)
2119, 20syl6ibr 240 . . . . . . . . . . 11 𝑋𝑁 → ((𝐴𝑥) ∈ 𝑁 → (𝐴𝑥) ≠ 𝑋))
2216, 21sylbi 205 . . . . . . . . . 10 (𝑋𝑁 → ((𝐴𝑥) ∈ 𝑁 → (𝐴𝑥) ≠ 𝑋))
2315, 22syl 17 . . . . . . . . 9 (𝜑 → ((𝐴𝑥) ∈ 𝑁 → (𝐴𝑥) ≠ 𝑋))
2423adantr 479 . . . . . . . 8 ((𝜑𝑥𝐷) → ((𝐴𝑥) ∈ 𝑁 → (𝐴𝑥) ≠ 𝑋))
2524com12 32 . . . . . . 7 ((𝐴𝑥) ∈ 𝑁 → ((𝜑𝑥𝐷) → (𝐴𝑥) ≠ 𝑋))
2625adantl 480 . . . . . 6 (((𝐴𝑥) ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∧ (𝐴𝑥) ∈ 𝑁) → ((𝜑𝑥𝐷) → (𝐴𝑥) ≠ 𝑋))
2714, 26sylbi 205 . . . . 5 ((𝐴𝑥) ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁) → ((𝜑𝑥𝐷) → (𝐴𝑥) ≠ 𝑋))
2813, 27syl 17 . . . 4 (({(𝐴𝑥)} = ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁) ∧ (𝐴𝑥) ∈ {(𝐴𝑥)}) → ((𝜑𝑥𝐷) → (𝐴𝑥) ≠ 𝑋))
2911, 28mpan2 702 . . 3 ({(𝐴𝑥)} = ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁) → ((𝜑𝑥𝐷) → (𝐴𝑥) ≠ 𝑋))
309, 29mpcom 37 . 2 ((𝜑𝑥𝐷) → (𝐴𝑥) ≠ 𝑋)
3130ralrimiva 2948 1 (𝜑 → ∀𝑥𝐷 (𝐴𝑥) ≠ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1976  wne 2779  wnel 2780  wral 2895  cin 3538  {csn 4124  {cpr 4126  cop 4130   class class class wbr 4577  cmpt 4637  ran crn 5029  cfv 5790  crio 6488  (class class class)co 6527   Neighbors cnbgra 25740   FriendGrph cfrgra 26309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-er 7607  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-card 8626  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-nn 10871  df-2 10929  df-n0 11143  df-z 11214  df-uz 11523  df-fz 12156  df-hash 12938  df-usgra 25656  df-nbgra 25743  df-frgra 26310
This theorem is referenced by: (None)
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