Theorem cplgredgex 32982

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 1135	. . 3 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐴 ∈ 𝑉)
2		simp3 1136	. . 3 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐵 ∈ (𝑉 ∖ {𝐴}))
3		eleq1 2826	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ 𝑉 ↔ 𝐴 ∈ 𝑉))
4		sneq 4568	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
5	4	difeq2d 4053	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝐴}))
6	5	eleq2d 2824	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑏 ∈ (𝑉 ∖ {𝑎}) ↔ 𝑏 ∈ (𝑉 ∖ {𝐴})))
7	3, 6	anbi12d 630	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ↔ (𝐴 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝐴}))))
8		preq1 4666	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏})
9	8	sseq1d 3948	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝐴, 𝑏} ⊆ 𝑒))
10	9	rexbidv 3225	. . . . . 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 2826	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ (𝑉 ∖ {𝐴}) ↔ 𝐵 ∈ (𝑉 ∖ {𝐴})))
13	12	anbi2d 628	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝐴})) ↔ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))))
14		preq2 4667	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵})
15	14	sseq1d 3948	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ({𝐴, 𝑏} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ 𝑒))
16	15	rexbidv 3225	. . . . . 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 27687	. . . . . . 7 ⊢ (𝐺 ∈ ComplGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒))
22	21	ibi 266	. . . . . 6 ⊢ (𝐺 ∈ ComplGraph → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒)
23		rsp2 3136	. . . . . 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 1131	. . . 4 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒))
26	1, 2, 18, 25	vtocl2d 3486	. . 3 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒))
27	1, 2, 26	mp2and 695	. 2 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒)
28	27	3expib 1120	1 ⊢ (𝐺 ∈ ComplGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ∖ cdif 3880   ⊆ wss 3883  {csn 4558  {cpr 4560  ‘cfv 6418  Vtxcvtx 27269  Edgcedg 27320  ComplGraphccplgr 27679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-nbgr 27603  df-uvtx 27656  df-cplgr 27681

This theorem is referenced by:  cusgredgex  32983