Theorem cplgredgex 35508

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 1153	. . 3 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐴 ∈ 𝑉)
2		simp3 1154	. . 3 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐵 ∈ (𝑉 ∖ {𝐴}))
3		eleq1 2857	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ 𝑉 ↔ 𝐴 ∈ 𝑉))
4		sneq 4601	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
5	4	difeq2d 4089	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝐴}))
6	5	eleq2d 2855	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑏 ∈ (𝑉 ∖ {𝑎}) ↔ 𝑏 ∈ (𝑉 ∖ {𝐴})))
7	3, 6	anbi12d 643	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ↔ (𝐴 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝐴}))))
8		preq1 4701	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏})
9	8	sseq1d 3976	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝐴, 𝑏} ⊆ 𝑒))
10	9	rexbidv 3195	. . . . . 6 ⊢ (𝑎 = 𝐴 → (∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝐴, 𝑏} ⊆ 𝑒))
11	7, 10	imbi12d 347	. . . . 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 2857	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ (𝑉 ∖ {𝐴}) ↔ 𝐵 ∈ (𝑉 ∖ {𝐴})))
13	12	anbi2d 641	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝐴})) ↔ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))))
14		preq2 4702	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵})
15	14	sseq1d 3976	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ({𝐴, 𝑏} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ 𝑒))
16	15	rexbidv 3195	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∃𝑒 ∈ 𝐸 {𝐴, 𝑏} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒))
17	13, 16	imbi12d 347	. . . . 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 518	. . . 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 29704	. . . . . . 7 ⊢ (𝐺 ∈ ComplGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒))
22	21	ibi 270	. . . . . 6 ⊢ (𝐺 ∈ ComplGraph → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒)
23		rsp2 3288	. . . . . 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 18	. . . . 5 ⊢ (𝐺 ∈ ComplGraph → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒))
25	24	3ad2ant1 1149	. . . 4 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒))
26	1, 2, 18, 25	vtocl2d 3537	. . 3 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒))
27	1, 2, 26	mp2and 711	. 2 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒)
28	27	3expib 1138	1 ⊢ (𝐺 ∈ ComplGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095   ∖ cdif 3910   ⊆ wss 3913  {csn 4591  {cpr 4593  ‘cfv 6533  Vtxcvtx 29283  Edgcedg 29334  ComplGraphccplgr 29696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-nbgr 29620  df-uvtx 29673  df-cplgr 29698

This theorem is referenced by:  cusgredgex  35509