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Theorem frgranbnb 26547
 Description: If two neighbors of a specific vertex have a common neighbor in a friendship graph, then this common neighbor must be the specific vertex. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
Hypotheses
Ref Expression
frgranbnb.x (𝜑𝑋𝑉)
frgranbnb.nx 𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)
frgranbnb.f (𝜑𝑉 FriendGrph 𝐸)
Assertion
Ref Expression
frgranbnb ((𝜑 ∧ (𝑈𝐷𝑊𝐷) ∧ 𝑈𝑊) → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → 𝐴 = 𝑋))

Proof of Theorem frgranbnb
StepHypRef Expression
1 frgranbnb.f . . . 4 (𝜑𝑉 FriendGrph 𝐸)
2 frisusgra 26519 . . . 4 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
31, 2syl 17 . . 3 (𝜑𝑉 USGrph 𝐸)
4 frgranbnb.nx . . . . . . . . . 10 𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)
54eleq2i 2680 . . . . . . . . 9 (𝑈𝐷𝑈 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))
6 nbgraeledg 25959 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → (𝑈 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) ↔ {𝑈, 𝑋} ∈ ran 𝐸))
76biimpd 218 . . . . . . . . 9 (𝑉 USGrph 𝐸 → (𝑈 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) → {𝑈, 𝑋} ∈ ran 𝐸))
85, 7syl5bi 231 . . . . . . . 8 (𝑉 USGrph 𝐸 → (𝑈𝐷 → {𝑈, 𝑋} ∈ ran 𝐸))
94eleq2i 2680 . . . . . . . . 9 (𝑊𝐷𝑊 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))
10 nbgraeledg 25959 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → (𝑊 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) ↔ {𝑊, 𝑋} ∈ ran 𝐸))
1110biimpd 218 . . . . . . . . 9 (𝑉 USGrph 𝐸 → (𝑊 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) → {𝑊, 𝑋} ∈ ran 𝐸))
129, 11syl5bi 231 . . . . . . . 8 (𝑉 USGrph 𝐸 → (𝑊𝐷 → {𝑊, 𝑋} ∈ ran 𝐸))
138, 12anim12d 584 . . . . . . 7 (𝑉 USGrph 𝐸 → ((𝑈𝐷𝑊𝐷) → ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)))
1413imp 444 . . . . . 6 ((𝑉 USGrph 𝐸 ∧ (𝑈𝐷𝑊𝐷)) → ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸))
15 nbgraisvtx 25960 . . . . . . . . 9 (𝑉 USGrph 𝐸 → (𝑈 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) → 𝑈𝑉))
165, 15syl5bi 231 . . . . . . . 8 (𝑉 USGrph 𝐸 → (𝑈𝐷𝑈𝑉))
17 nbgraisvtx 25960 . . . . . . . . 9 (𝑉 USGrph 𝐸 → (𝑊 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) → 𝑊𝑉))
189, 17syl5bi 231 . . . . . . . 8 (𝑉 USGrph 𝐸 → (𝑊𝐷𝑊𝑉))
1916, 18anim12d 584 . . . . . . 7 (𝑉 USGrph 𝐸 → ((𝑈𝐷𝑊𝐷) → (𝑈𝑉𝑊𝑉)))
2019imp 444 . . . . . 6 ((𝑉 USGrph 𝐸 ∧ (𝑈𝐷𝑊𝐷)) → (𝑈𝑉𝑊𝑉))
21 usgraedgrnv 25906 . . . . . . . . . . . . . 14 ((𝑉 USGrph 𝐸 ∧ {𝑈, 𝐴} ∈ ran 𝐸) → (𝑈𝑉𝐴𝑉))
2221adantrr 749 . . . . . . . . . . . . 13 ((𝑉 USGrph 𝐸 ∧ ({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸)) → (𝑈𝑉𝐴𝑉))
23 frgranbnb.x . . . . . . . . . . . . . . . . . 18 (𝜑𝑋𝑉)
24 ax-1 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴 = 𝑋 → (𝑉 FriendGrph 𝐸𝐴 = 𝑋))
25242a1d 26 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴 = 𝑋 → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → (((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉)) → (𝑉 FriendGrph 𝐸𝐴 = 𝑋))))
26252a1d 26 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 = 𝑋 → (𝑈𝑊 → (({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸) → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → (((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉)) → (𝑉 FriendGrph 𝐸𝐴 = 𝑋))))))
27 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) → 𝑉 USGrph 𝐸)
2827adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ (𝐴𝑋𝑈𝑊)) → 𝑉 USGrph 𝐸)
29 simprrr 801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) → 𝑊𝑉)
3029adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ (𝐴𝑋𝑈𝑊)) → 𝑊𝑉)
31 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((𝑈𝑉𝑊𝑉) → 𝑈𝑉)
3231adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉)) → 𝑈𝑉)
3332adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) → 𝑈𝑉)
3433adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ (𝐴𝑋𝑈𝑊)) → 𝑈𝑉)
35 necom 2835 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑈𝑊𝑊𝑈)
3635biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑈𝑊𝑊𝑈)
3736adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝐴𝑋𝑈𝑊) → 𝑊𝑈)
3837adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ (𝐴𝑋𝑈𝑊)) → 𝑊𝑈)
3930, 34, 383jca 1235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ (𝐴𝑋𝑈𝑊)) → (𝑊𝑉𝑈𝑉𝑊𝑈))
40 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((𝑋𝑉𝐴𝑉) → 𝑋𝑉)
4140adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉)) → 𝑋𝑉)
4241adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) → 𝑋𝑉)
4342adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ (𝐴𝑋𝑈𝑊)) → 𝑋𝑉)
44 simprlr 799 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) → 𝐴𝑉)
4544adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ (𝐴𝑋𝑈𝑊)) → 𝐴𝑉)
46 necom 2835 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝐴𝑋𝑋𝐴)
4746biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝐴𝑋𝑋𝐴)
4847adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝐴𝑋𝑈𝑊) → 𝑋𝐴)
4948adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ (𝐴𝑋𝑈𝑊)) → 𝑋𝐴)
5043, 45, 493jca 1235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ (𝐴𝑋𝑈𝑊)) → (𝑋𝑉𝐴𝑉𝑋𝐴))
5128, 39, 503jca 1235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ (𝐴𝑋𝑈𝑊)) → (𝑉 USGrph 𝐸 ∧ (𝑊𝑉𝑈𝑉𝑊𝑈) ∧ (𝑋𝑉𝐴𝑉𝑋𝐴)))
5251ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) → ((𝐴𝑋𝑈𝑊) → (𝑉 USGrph 𝐸 ∧ (𝑊𝑉𝑈𝑉𝑊𝑈) ∧ (𝑋𝑉𝐴𝑉𝑋𝐴))))
5352adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) → ((𝐴𝑋𝑈𝑊) → (𝑉 USGrph 𝐸 ∧ (𝑊𝑉𝑈𝑉𝑊𝑈) ∧ (𝑋𝑉𝐴𝑉𝑋𝐴))))
5453adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) ∧ ({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸)) → ((𝐴𝑋𝑈𝑊) → (𝑉 USGrph 𝐸 ∧ (𝑊𝑉𝑈𝑉𝑊𝑈) ∧ (𝑋𝑉𝐴𝑉𝑋𝐴))))
5554imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) ∧ ({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸)) ∧ (𝐴𝑋𝑈𝑊)) → (𝑉 USGrph 𝐸 ∧ (𝑊𝑉𝑈𝑉𝑊𝑈) ∧ (𝑋𝑉𝐴𝑉𝑋𝐴)))
56 prcom 4211 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 {𝑈, 𝑋} = {𝑋, 𝑈}
5756eleq1i 2679 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ({𝑈, 𝑋} ∈ ran 𝐸 ↔ {𝑋, 𝑈} ∈ ran 𝐸)
5857biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ({𝑈, 𝑋} ∈ ran 𝐸 → {𝑋, 𝑈} ∈ ran 𝐸)
5958anim1i 590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸) → ({𝑋, 𝑈} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸))
6059ancomd 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸) → ({𝑊, 𝑋} ∈ ran 𝐸 ∧ {𝑋, 𝑈} ∈ ran 𝐸))
6160adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) → ({𝑊, 𝑋} ∈ ran 𝐸 ∧ {𝑋, 𝑈} ∈ ran 𝐸))
62 prcom 4211 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 {𝑊, 𝐴} = {𝐴, 𝑊}
6362eleq1i 2679 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ({𝑊, 𝐴} ∈ ran 𝐸 ↔ {𝐴, 𝑊} ∈ ran 𝐸)
6463biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ({𝑊, 𝐴} ∈ ran 𝐸 → {𝐴, 𝑊} ∈ ran 𝐸)
6564anim2i 591 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → ({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝑊} ∈ ran 𝐸))
6661, 65anim12i 588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) ∧ ({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸)) → (({𝑊, 𝑋} ∈ ran 𝐸 ∧ {𝑋, 𝑈} ∈ ran 𝐸) ∧ ({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝑊} ∈ ran 𝐸)))
6766adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) ∧ ({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸)) ∧ (𝐴𝑋𝑈𝑊)) → (({𝑊, 𝑋} ∈ ran 𝐸 ∧ {𝑋, 𝑈} ∈ ran 𝐸) ∧ ({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝑊} ∈ ran 𝐸)))
68 4cyclusnfrgra 26546 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑉 USGrph 𝐸 ∧ (𝑊𝑉𝑈𝑉𝑊𝑈) ∧ (𝑋𝑉𝐴𝑉𝑋𝐴)) → ((({𝑊, 𝑋} ∈ ran 𝐸 ∧ {𝑋, 𝑈} ∈ ran 𝐸) ∧ ({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝑊} ∈ ran 𝐸)) → ¬ 𝑉 FriendGrph 𝐸))
6955, 67, 68sylc 63 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) ∧ ({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸)) ∧ (𝐴𝑋𝑈𝑊)) → ¬ 𝑉 FriendGrph 𝐸)
7069pm2.21d 117 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) ∧ ({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸)) ∧ (𝐴𝑋𝑈𝑊)) → (𝑉 FriendGrph 𝐸𝐴 = 𝑋))
7170ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) ∧ ({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸)) → ((𝐴𝑋𝑈𝑊) → (𝑉 FriendGrph 𝐸𝐴 = 𝑋)))
7271com23 84 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑉 USGrph 𝐸 ∧ ((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉))) ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) ∧ ({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸)) → (𝑉 FriendGrph 𝐸 → ((𝐴𝑋𝑈𝑊) → 𝐴 = 𝑋)))
7372exp41 636 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑉 USGrph 𝐸 → (((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉)) → (({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸) → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → (𝑉 FriendGrph 𝐸 → ((𝐴𝑋𝑈𝑊) → 𝐴 = 𝑋))))))
7473com25 97 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑉 USGrph 𝐸 → (𝑉 FriendGrph 𝐸 → (({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸) → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → (((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉)) → ((𝐴𝑋𝑈𝑊) → 𝐴 = 𝑋))))))
752, 74mpcom 37 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑉 FriendGrph 𝐸 → (({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸) → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → (((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉)) → ((𝐴𝑋𝑈𝑊) → 𝐴 = 𝑋)))))
7675com15 99 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴𝑋𝑈𝑊) → (({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸) → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → (((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉)) → (𝑉 FriendGrph 𝐸𝐴 = 𝑋)))))
7776ex 449 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴𝑋 → (𝑈𝑊 → (({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸) → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → (((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉)) → (𝑉 FriendGrph 𝐸𝐴 = 𝑋))))))
7826, 77pm2.61ine 2865 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑈𝑊 → (({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸) → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → (((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉)) → (𝑉 FriendGrph 𝐸𝐴 = 𝑋)))))
7978imp 444 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑈𝑊 ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → (((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉)) → (𝑉 FriendGrph 𝐸𝐴 = 𝑋))))
8079com13 86 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑋𝑉𝐴𝑉) ∧ (𝑈𝑉𝑊𝑉)) → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → ((𝑈𝑊 ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) → (𝑉 FriendGrph 𝐸𝐴 = 𝑋))))
8180ex 449 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋𝑉𝐴𝑉) → ((𝑈𝑉𝑊𝑉) → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → ((𝑈𝑊 ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) → (𝑉 FriendGrph 𝐸𝐴 = 𝑋)))))
8281com25 97 . . . . . . . . . . . . . . . . . . . 20 ((𝑋𝑉𝐴𝑉) → (𝑉 FriendGrph 𝐸 → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → ((𝑈𝑊 ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) → ((𝑈𝑉𝑊𝑉) → 𝐴 = 𝑋)))))
8382ex 449 . . . . . . . . . . . . . . . . . . 19 (𝑋𝑉 → (𝐴𝑉 → (𝑉 FriendGrph 𝐸 → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → ((𝑈𝑊 ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) → ((𝑈𝑉𝑊𝑉) → 𝐴 = 𝑋))))))
8483com23 84 . . . . . . . . . . . . . . . . . 18 (𝑋𝑉 → (𝑉 FriendGrph 𝐸 → (𝐴𝑉 → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → ((𝑈𝑊 ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) → ((𝑈𝑉𝑊𝑉) → 𝐴 = 𝑋))))))
8523, 1, 84sylc 63 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐴𝑉 → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → ((𝑈𝑊 ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) → ((𝑈𝑉𝑊𝑉) → 𝐴 = 𝑋)))))
8685com13 86 . . . . . . . . . . . . . . . 16 (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → (𝐴𝑉 → (𝜑 → ((𝑈𝑊 ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) → ((𝑈𝑉𝑊𝑉) → 𝐴 = 𝑋)))))
8786adantl 481 . . . . . . . . . . . . . . 15 ((𝑉 USGrph 𝐸 ∧ ({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸)) → (𝐴𝑉 → (𝜑 → ((𝑈𝑊 ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) → ((𝑈𝑉𝑊𝑉) → 𝐴 = 𝑋)))))
8887com12 32 . . . . . . . . . . . . . 14 (𝐴𝑉 → ((𝑉 USGrph 𝐸 ∧ ({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸)) → (𝜑 → ((𝑈𝑊 ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) → ((𝑈𝑉𝑊𝑉) → 𝐴 = 𝑋)))))
8988adantl 481 . . . . . . . . . . . . 13 ((𝑈𝑉𝐴𝑉) → ((𝑉 USGrph 𝐸 ∧ ({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸)) → (𝜑 → ((𝑈𝑊 ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) → ((𝑈𝑉𝑊𝑉) → 𝐴 = 𝑋)))))
9022, 89mpcom 37 . . . . . . . . . . . 12 ((𝑉 USGrph 𝐸 ∧ ({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸)) → (𝜑 → ((𝑈𝑊 ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) → ((𝑈𝑉𝑊𝑉) → 𝐴 = 𝑋))))
9190ex 449 . . . . . . . . . . 11 (𝑉 USGrph 𝐸 → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → (𝜑 → ((𝑈𝑊 ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) → ((𝑈𝑉𝑊𝑉) → 𝐴 = 𝑋)))))
9291com25 97 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → ((𝑈𝑉𝑊𝑉) → (𝜑 → ((𝑈𝑊 ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → 𝐴 = 𝑋)))))
9392com14 94 . . . . . . . . 9 ((𝑈𝑊 ∧ ({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸)) → ((𝑈𝑉𝑊𝑉) → (𝜑 → (𝑉 USGrph 𝐸 → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → 𝐴 = 𝑋)))))
9493ex 449 . . . . . . . 8 (𝑈𝑊 → (({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸) → ((𝑈𝑉𝑊𝑉) → (𝜑 → (𝑉 USGrph 𝐸 → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → 𝐴 = 𝑋))))))
9594com15 99 . . . . . . 7 (𝑉 USGrph 𝐸 → (({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸) → ((𝑈𝑉𝑊𝑉) → (𝜑 → (𝑈𝑊 → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → 𝐴 = 𝑋))))))
9695adantr 480 . . . . . 6 ((𝑉 USGrph 𝐸 ∧ (𝑈𝐷𝑊𝐷)) → (({𝑈, 𝑋} ∈ ran 𝐸 ∧ {𝑊, 𝑋} ∈ ran 𝐸) → ((𝑈𝑉𝑊𝑉) → (𝜑 → (𝑈𝑊 → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → 𝐴 = 𝑋))))))
9714, 20, 96mp2d 47 . . . . 5 ((𝑉 USGrph 𝐸 ∧ (𝑈𝐷𝑊𝐷)) → (𝜑 → (𝑈𝑊 → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → 𝐴 = 𝑋))))
9897ex 449 . . . 4 (𝑉 USGrph 𝐸 → ((𝑈𝐷𝑊𝐷) → (𝜑 → (𝑈𝑊 → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → 𝐴 = 𝑋)))))
9998com23 84 . . 3 (𝑉 USGrph 𝐸 → (𝜑 → ((𝑈𝐷𝑊𝐷) → (𝑈𝑊 → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → 𝐴 = 𝑋)))))
1003, 99mpcom 37 . 2 (𝜑 → ((𝑈𝐷𝑊𝐷) → (𝑈𝑊 → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → 𝐴 = 𝑋))))
1011003imp 1249 1 ((𝜑 ∧ (𝑈𝐷𝑊𝐷) ∧ 𝑈𝑊) → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → 𝐴 = 𝑋))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {cpr 4127  ⟨cop 4131   class class class wbr 4583  ran crn 5039  (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-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-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:  frgrancvvdeqlemB  26565
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