Theorem cplgredgex 35108

Description: Any two (distinct) vertices in a complete graph are connected to each other by at least one edge. (Contributed by BTernaryTau, 2-Oct-2023.)

Hypotheses

Ref	Expression
cplgredgex.1	⊢ 𝑉 = (Vtx‘𝐺)
cplgredgex.2	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
cplgredgex	⊢ (𝐺 ∈ ComplGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒))

Distinct variable groups:   𝐴,𝑒   𝐵,𝑒   𝑒,𝐸   𝑒,𝐺   𝑒,𝑉

Proof of Theorem cplgredgex

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1137	. . 3 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐴 ∈ 𝑉)
2		simp3 1138	. . 3 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐵 ∈ (𝑉 ∖ {𝐴}))
3		eleq1 2816	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ 𝑉 ↔ 𝐴 ∈ 𝑉))
4		sneq 4599	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
5	4	difeq2d 4089	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝐴}))
6	5	eleq2d 2814	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑏 ∈ (𝑉 ∖ {𝑎}) ↔ 𝑏 ∈ (𝑉 ∖ {𝐴})))
7	3, 6	anbi12d 632	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ↔ (𝐴 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝐴}))))
8		preq1 4697	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏})
9	8	sseq1d 3978	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝐴, 𝑏} ⊆ 𝑒))
10	9	rexbidv 3157	. . . . . 6 ⊢ (𝑎 = 𝐴 → (∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝐴, 𝑏} ⊆ 𝑒))
11	7, 10	imbi12d 344	. . . . 5 ⊢ (𝑎 = 𝐴 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒) ↔ ((𝐴 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝑏} ⊆ 𝑒)))
12		eleq1 2816	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ (𝑉 ∖ {𝐴}) ↔ 𝐵 ∈ (𝑉 ∖ {𝐴})))
13	12	anbi2d 630	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝐴})) ↔ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))))
14		preq2 4698	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵})
15	14	sseq1d 3978	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ({𝐴, 𝑏} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ 𝑒))
16	15	rexbidv 3157	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∃𝑒 ∈ 𝐸 {𝐴, 𝑏} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒))
17	13, 16	imbi12d 344	. . . . 5 ⊢ (𝑏 = 𝐵 → (((𝐴 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝑏} ⊆ 𝑒) ↔ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒)))
18	11, 17	sylan9bb 509	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒) ↔ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒)))
19		cplgredgex.1	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
20		cplgredgex.2	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
21	19, 20	iscplgredg 29344	. . . . . . 7 ⊢ (𝐺 ∈ ComplGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒))
22	21	ibi 267	. . . . . 6 ⊢ (𝐺 ∈ ComplGraph → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒)
23		rsp2 3254	. . . . . 6 ⊢ (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒 → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒))
24	22, 23	syl 17	. . . . 5 ⊢ (𝐺 ∈ ComplGraph → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒))
25	24	3ad2ant1 1133	. . . 4 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒))
26	1, 2, 18, 25	vtocl2d 3528	. . 3 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒))
27	1, 2, 26	mp2and 699	. 2 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒)
28	27	3expib 1122	1 ⊢ (𝐺 ∈ ComplGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ∖ cdif 3911   ⊆ wss 3914  {csn 4589  {cpr 4591  ‘cfv 6511  Vtxcvtx 28923  Edgcedg 28974  ComplGraphccplgr 29336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-nbgr 29260  df-uvtx 29313  df-cplgr 29338

This theorem is referenced by:  cusgredgex  35109