Theorem frgraeu 26581

Description: Any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 18-Feb-2018.)

Assertion

Ref	Expression
frgraeu	⊢ (𝑉 FriendGrph 𝐸 → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)))

Distinct variable groups:   𝐴,𝑏   𝐶,𝑏   𝐸,𝑏   𝑉,𝑏

Proof of Theorem frgraeu

Step	Hyp	Ref	Expression
1		frgraun 26523	. . . 4 ⊢ (𝑉 FriendGrph 𝐸 → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)))
2	1	imp 444	. . 3 ⊢ ((𝑉 FriendGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → ∃!𝑏 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸))
3		df-reu 2903	. . . 4 ⊢ (∃!𝑏 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸) ↔ ∃!𝑏(𝑏 ∈ 𝑉 ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)))
4		frisusgra 26519	. . . . . . . . . 10 ⊢ (𝑉 FriendGrph 𝐸 → 𝑉 USGrph 𝐸)
5		usgraedgrnv 25906	. . . . . . . . . . . . 13 ⊢ ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝑏} ∈ ran 𝐸) → (𝐴 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))
6	5	simprd 478	. . . . . . . . . . . 12 ⊢ ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝑏} ∈ ran 𝐸) → 𝑏 ∈ 𝑉)
7	6	expcom 450	. . . . . . . . . . 11 ⊢ ({𝐴, 𝑏} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → 𝑏 ∈ 𝑉))
8	7	adantr 480	. . . . . . . . . 10 ⊢ (({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸) → (𝑉 USGrph 𝐸 → 𝑏 ∈ 𝑉))
9	4, 8	syl5com 31	. . . . . . . . 9 ⊢ (𝑉 FriendGrph 𝐸 → (({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸) → 𝑏 ∈ 𝑉))
10	9	adantr 480	. . . . . . . 8 ⊢ ((𝑉 FriendGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → (({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸) → 𝑏 ∈ 𝑉))
11	10	pm4.71rd 665	. . . . . . 7 ⊢ ((𝑉 FriendGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → (({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸) ↔ (𝑏 ∈ 𝑉 ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸))))
12	11	bicomd 212	. . . . . 6 ⊢ ((𝑉 FriendGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → ((𝑏 ∈ 𝑉 ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)) ↔ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)))
13	12	eubidv 2478	. . . . 5 ⊢ ((𝑉 FriendGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → (∃!𝑏(𝑏 ∈ 𝑉 ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)) ↔ ∃!𝑏({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)))
14	13	biimpcd 238	. . . 4 ⊢ (∃!𝑏(𝑏 ∈ 𝑉 ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)) → ((𝑉 FriendGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → ∃!𝑏({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)))
15	3, 14	sylbi 206	. . 3 ⊢ (∃!𝑏 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸) → ((𝑉 FriendGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → ∃!𝑏({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)))
16	2, 15	mpcom 37	. 2 ⊢ ((𝑉 FriendGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → ∃!𝑏({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸))
17	16	ex 449	1 ⊢ (𝑉 FriendGrph 𝐸 → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977  ∃!weu 2458   ≠ wne 2780  ∃!wreu 2898  {cpr 4127   class class class wbr 4583  ran crn 5039   USGrph cusg 25859   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-frgra 26516

This theorem is referenced by:  frg2woteqm  26586