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Theorem frgrawopreg 26576
 Description: In a friendship graph there are either no vertices or exactly one vertex having degree K, or all or all except one vertices have degree K. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
Hypotheses
Ref Expression
frgrawopreg.a 𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
frgrawopreg.b 𝐵 = (𝑉𝐴)
Assertion
Ref Expression
frgrawopreg (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐸   𝑥,𝐾   𝑥,𝑉   𝑥,𝐵

Proof of Theorem frgrawopreg
Dummy variables 𝑏 𝑦 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrawopreg.a . . 3 𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
2 frgrawopreg.b . . 3 𝐵 = (𝑉𝐴)
31, 2frgrawopreglem1 26571 . 2 (𝑉 FriendGrph 𝐸 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4 hashv01gt1 12995 . . . 4 (𝐴 ∈ V → ((#‘𝐴) = 0 ∨ (#‘𝐴) = 1 ∨ 1 < (#‘𝐴)))
5 hashv01gt1 12995 . . . 4 (𝐵 ∈ V → ((#‘𝐵) = 0 ∨ (#‘𝐵) = 1 ∨ 1 < (#‘𝐵)))
64, 5anim12i 588 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (((#‘𝐴) = 0 ∨ (#‘𝐴) = 1 ∨ 1 < (#‘𝐴)) ∧ ((#‘𝐵) = 0 ∨ (#‘𝐵) = 1 ∨ 1 < (#‘𝐵))))
7 hasheq0 13015 . . . . . . . . . . . . 13 (𝐴 ∈ V → ((#‘𝐴) = 0 ↔ 𝐴 = ∅))
87biimpd 218 . . . . . . . . . . . 12 (𝐴 ∈ V → ((#‘𝐴) = 0 → 𝐴 = ∅))
98adantr 480 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((#‘𝐴) = 0 → 𝐴 = ∅))
109impcom 445 . . . . . . . . . 10 (((#‘𝐴) = 0 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → 𝐴 = ∅)
1110olcd 407 . . . . . . . . 9 (((#‘𝐴) = 0 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → ((#‘𝐴) = 1 ∨ 𝐴 = ∅))
1211orcd 406 . . . . . . . 8 (((#‘𝐴) = 0 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))
1312a1d 25 . . . . . . 7 (((#‘𝐴) = 0 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
1413ex 449 . . . . . 6 ((#‘𝐴) = 0 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))))
1514a1d 25 . . . . 5 ((#‘𝐴) = 0 → (((#‘𝐵) = 0 ∨ (#‘𝐵) = 1 ∨ 1 < (#‘𝐵)) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))))
16 orc 399 . . . . . . . . 9 ((#‘𝐴) = 1 → ((#‘𝐴) = 1 ∨ 𝐴 = ∅))
1716orcd 406 . . . . . . . 8 ((#‘𝐴) = 1 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))
1817a1d 25 . . . . . . 7 ((#‘𝐴) = 1 → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
1918a1d 25 . . . . . 6 ((#‘𝐴) = 1 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))))
2019a1d 25 . . . . 5 ((#‘𝐴) = 1 → (((#‘𝐵) = 0 ∨ (#‘𝐵) = 1 ∨ 1 < (#‘𝐵)) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))))
21 hasheq0 13015 . . . . . . . . . . . . . . 15 (𝐵 ∈ V → ((#‘𝐵) = 0 ↔ 𝐵 = ∅))
2221biimpd 218 . . . . . . . . . . . . . 14 (𝐵 ∈ V → ((#‘𝐵) = 0 → 𝐵 = ∅))
2322adantl 481 . . . . . . . . . . . . 13 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((#‘𝐵) = 0 → 𝐵 = ∅))
2423impcom 445 . . . . . . . . . . . 12 (((#‘𝐵) = 0 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → 𝐵 = ∅)
2524olcd 407 . . . . . . . . . . 11 (((#‘𝐵) = 0 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → ((#‘𝐵) = 1 ∨ 𝐵 = ∅))
2625olcd 407 . . . . . . . . . 10 (((#‘𝐵) = 0 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))
2726a1d 25 . . . . . . . . 9 (((#‘𝐵) = 0 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
2827ex 449 . . . . . . . 8 ((#‘𝐵) = 0 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))))
2928a1d 25 . . . . . . 7 ((#‘𝐵) = 0 → (1 < (#‘𝐴) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))))
30 orc 399 . . . . . . . . . . 11 ((#‘𝐵) = 1 → ((#‘𝐵) = 1 ∨ 𝐵 = ∅))
3130olcd 407 . . . . . . . . . 10 ((#‘𝐵) = 1 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))
3231a1d 25 . . . . . . . . 9 ((#‘𝐵) = 1 → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
3332a1d 25 . . . . . . . 8 ((#‘𝐵) = 1 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))))
3433a1d 25 . . . . . . 7 ((#‘𝐵) = 1 → (1 < (#‘𝐴) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))))
351, 2frgrawopreglem5 26575 . . . . . . . . . . . 12 ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝐴) ∧ 1 < (#‘𝐵)) → ∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)))
36353expb 1258 . . . . . . . . . . 11 ((𝑉 FriendGrph 𝐸 ∧ (1 < (#‘𝐴) ∧ 1 < (#‘𝐵))) → ∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)))
37 frisusgra 26519 . . . . . . . . . . . . . . . 16 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
38 simplll 794 . . . . . . . . . . . . . . . . . . . 20 ((((𝑉 USGrph 𝐸 ∧ ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) → 𝑉 USGrph 𝐸)
39 elrabi 3328 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 ∈ {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾} → 𝑎𝑉)
4039, 1eleq2s 2706 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎𝐴𝑎𝑉)
4140ad2antrl 760 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑉 USGrph 𝐸 ∧ ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎𝐴𝑥𝐴)) → 𝑎𝑉)
4241adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑉 USGrph 𝐸 ∧ ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) → 𝑎𝑉)
431rabeq2i 3170 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥𝐴 ↔ (𝑥𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾))
4443simplbi 475 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥𝐴𝑥𝑉)
4544ad2antll 761 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑉 USGrph 𝐸 ∧ ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎𝐴𝑥𝐴)) → 𝑥𝑉)
4645adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑉 USGrph 𝐸 ∧ ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) → 𝑥𝑉)
47 simpr1r 1112 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑉 USGrph 𝐸 ∧ ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) → 𝑎𝑥)
4847adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑉 USGrph 𝐸 ∧ ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎𝐴𝑥𝐴)) → 𝑎𝑥)
4948adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑉 USGrph 𝐸 ∧ ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) → 𝑎𝑥)
5042, 46, 493jca 1235 . . . . . . . . . . . . . . . . . . . 20 ((((𝑉 USGrph 𝐸 ∧ ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) → (𝑎𝑉𝑥𝑉𝑎𝑥))
512eleq2i 2680 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏𝐵𝑏 ∈ (𝑉𝐴))
52 eldif 3550 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 ∈ (𝑉𝐴) ↔ (𝑏𝑉 ∧ ¬ 𝑏𝐴))
5351, 52bitri 263 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏𝐵 ↔ (𝑏𝑉 ∧ ¬ 𝑏𝐴))
5453simplbi 475 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏𝐵𝑏𝑉)
5554ad2antrl 760 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑉 USGrph 𝐸 ∧ ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) → 𝑏𝑉)
562eleq2i 2680 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦𝐵𝑦 ∈ (𝑉𝐴))
57 eldif 3550 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (𝑉𝐴) ↔ (𝑦𝑉 ∧ ¬ 𝑦𝐴))
5856, 57bitri 263 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦𝐵 ↔ (𝑦𝑉 ∧ ¬ 𝑦𝐴))
5958simplbi 475 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝐵𝑦𝑉)
6059ad2antll 761 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑉 USGrph 𝐸 ∧ ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) → 𝑦𝑉)
61 simpr1l 1111 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑉 USGrph 𝐸 ∧ ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) → 𝑏𝑦)
6261adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑉 USGrph 𝐸 ∧ ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎𝐴𝑥𝐴)) → 𝑏𝑦)
6362adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑉 USGrph 𝐸 ∧ ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) → 𝑏𝑦)
6455, 60, 633jca 1235 . . . . . . . . . . . . . . . . . . . 20 ((((𝑉 USGrph 𝐸 ∧ ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) → (𝑏𝑉𝑦𝑉𝑏𝑦))
65 prcom 4211 . . . . . . . . . . . . . . . . . . . . . . . . . 26 {𝑥, 𝑏} = {𝑏, 𝑥}
6665eleq1i 2679 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({𝑥, 𝑏} ∈ ran 𝐸 ↔ {𝑏, 𝑥} ∈ ran 𝐸)
6766biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . 24 ({𝑥, 𝑏} ∈ ran 𝐸 → {𝑏, 𝑥} ∈ ran 𝐸)
6867anim2i 591 . . . . . . . . . . . . . . . . . . . . . . 23 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) → ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑥} ∈ ran 𝐸))
69 prcom 4211 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 {𝑎, 𝑦} = {𝑦, 𝑎}
7069eleq1i 2679 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({𝑎, 𝑦} ∈ ran 𝐸 ↔ {𝑦, 𝑎} ∈ ran 𝐸)
7170biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({𝑎, 𝑦} ∈ ran 𝐸 → {𝑦, 𝑎} ∈ ran 𝐸)
7271anim1i 590 . . . . . . . . . . . . . . . . . . . . . . . 24 (({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸) → ({𝑦, 𝑎} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))
7372ancomd 466 . . . . . . . . . . . . . . . . . . . . . . 23 (({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸) → ({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑎} ∈ ran 𝐸))
7468, 73anim12i 588 . . . . . . . . . . . . . . . . . . . . . 22 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)) → (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑥} ∈ ran 𝐸) ∧ ({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑎} ∈ ran 𝐸)))
75743adant1 1072 . . . . . . . . . . . . . . . . . . . . 21 (((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)) → (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑥} ∈ ran 𝐸) ∧ ({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑎} ∈ ran 𝐸)))
7675ad3antlr 763 . . . . . . . . . . . . . . . . . . . 20 ((((𝑉 USGrph 𝐸 ∧ ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) → (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑥} ∈ ran 𝐸) ∧ ({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑎} ∈ ran 𝐸)))
77 4cyclusnfrgra 26546 . . . . . . . . . . . . . . . . . . . . 21 ((𝑉 USGrph 𝐸 ∧ (𝑎𝑉𝑥𝑉𝑎𝑥) ∧ (𝑏𝑉𝑦𝑉𝑏𝑦)) → ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑥} ∈ ran 𝐸) ∧ ({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑎} ∈ ran 𝐸)) → ¬ 𝑉 FriendGrph 𝐸))
7877imp 444 . . . . . . . . . . . . . . . . . . . 20 (((𝑉 USGrph 𝐸 ∧ (𝑎𝑉𝑥𝑉𝑎𝑥) ∧ (𝑏𝑉𝑦𝑉𝑏𝑦)) ∧ (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑥} ∈ ran 𝐸) ∧ ({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑎} ∈ ran 𝐸))) → ¬ 𝑉 FriendGrph 𝐸)
7938, 50, 64, 76, 78syl31anc 1321 . . . . . . . . . . . . . . . . . . 19 ((((𝑉 USGrph 𝐸 ∧ ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) → ¬ 𝑉 FriendGrph 𝐸)
8079pm2.21d 117 . . . . . . . . . . . . . . . . . 18 ((((𝑉 USGrph 𝐸 ∧ ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸))) ∧ (𝑎𝐴𝑥𝐴)) ∧ (𝑏𝐵𝑦𝐵)) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
8180exp41 636 . . . . . . . . . . . . . . . . 17 (𝑉 USGrph 𝐸 → (((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)) → ((𝑎𝐴𝑥𝐴) → ((𝑏𝐵𝑦𝐵) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))))))
8281com25 97 . . . . . . . . . . . . . . . 16 (𝑉 USGrph 𝐸 → (𝑉 FriendGrph 𝐸 → ((𝑎𝐴𝑥𝐴) → ((𝑏𝐵𝑦𝐵) → (((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)) → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))))))
8337, 82mpcom 37 . . . . . . . . . . . . . . 15 (𝑉 FriendGrph 𝐸 → ((𝑎𝐴𝑥𝐴) → ((𝑏𝐵𝑦𝐵) → (((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)) → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))))
8483imp 444 . . . . . . . . . . . . . 14 ((𝑉 FriendGrph 𝐸 ∧ (𝑎𝐴𝑥𝐴)) → ((𝑏𝐵𝑦𝐵) → (((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)) → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))))
8584rexlimdvv 3019 . . . . . . . . . . . . 13 ((𝑉 FriendGrph 𝐸 ∧ (𝑎𝐴𝑥𝐴)) → (∃𝑏𝐵𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)) → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
8685rexlimdvva 3020 . . . . . . . . . . . 12 (𝑉 FriendGrph 𝐸 → (∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)) → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
8786adantr 480 . . . . . . . . . . 11 ((𝑉 FriendGrph 𝐸 ∧ (1 < (#‘𝐴) ∧ 1 < (#‘𝐵))) → (∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)) → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
8836, 87mpd 15 . . . . . . . . . 10 ((𝑉 FriendGrph 𝐸 ∧ (1 < (#‘𝐴) ∧ 1 < (#‘𝐵))) → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))
8988expcom 450 . . . . . . . . 9 ((1 < (#‘𝐴) ∧ 1 < (#‘𝐵)) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
9089a1d 25 . . . . . . . 8 ((1 < (#‘𝐴) ∧ 1 < (#‘𝐵)) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))))
9190expcom 450 . . . . . . 7 (1 < (#‘𝐵) → (1 < (#‘𝐴) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))))
9229, 34, 913jaoi 1383 . . . . . 6 (((#‘𝐵) = 0 ∨ (#‘𝐵) = 1 ∨ 1 < (#‘𝐵)) → (1 < (#‘𝐴) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))))
9392com12 32 . . . . 5 (1 < (#‘𝐴) → (((#‘𝐵) = 0 ∨ (#‘𝐵) = 1 ∨ 1 < (#‘𝐵)) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))))
9415, 20, 933jaoi 1383 . . . 4 (((#‘𝐴) = 0 ∨ (#‘𝐴) = 1 ∨ 1 < (#‘𝐴)) → (((#‘𝐵) = 0 ∨ (#‘𝐵) = 1 ∨ 1 < (#‘𝐵)) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))))
9594imp 444 . . 3 ((((#‘𝐴) = 0 ∨ (#‘𝐴) = 1 ∨ 1 < (#‘𝐴)) ∧ ((#‘𝐵) = 0 ∨ (#‘𝐵) = 1 ∨ 1 < (#‘𝐵))) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))))
966, 95mpcom 37 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))))
973, 96mpcom 37 1 (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  {cpr 4127   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   < clt 9953  #chash 12979   USGrph cusg 25859   VDeg cvdg 26420   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-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-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-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-hash 12980  df-usgra 25862  df-nbgra 25949  df-vdgr 26421  df-frgra 26516 This theorem is referenced by:  frgraregorufr0  26579
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