Theorem cplgredgex 36103

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 1155	. . 3 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐴 ∈ 𝑉)
2		simp3 1156	. . 3 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐵 ∈ (𝑉 ∖ {𝐴}))
3		eleq1 2852	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ 𝑉 ↔ 𝐴 ∈ 𝑉))
4		sneq 4594	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
5	4	difeq2d 4066	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝐴}))
6	5	eleq2d 2850	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑏 ∈ (𝑉 ∖ {𝑎}) ↔ 𝑏 ∈ (𝑉 ∖ {𝐴})))
7	3, 6	anbi12d 644	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) ↔ (𝐴 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝐴}))))
8		preq1 4696	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏})
9	8	sseq1d 3944	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝐴, 𝑏} ⊆ 𝑒))
10	9	rexbidv 3190	. . . . . 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 2852	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ (𝑉 ∖ {𝐴}) ↔ 𝐵 ∈ (𝑉 ∖ {𝐴})))
13	12	anbi2d 642	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝐴})) ↔ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴}))))
14		preq2 4697	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵})
15	14	sseq1d 3944	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ({𝐴, 𝑏} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ 𝑒))
16	15	rexbidv 3190	. . . . . 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 519	. . . 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 30055	. . . . . . 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 3283	. . . . . 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 1151	. . . 4 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒))
26	1, 2, 18, 25	vtocl2d 3527	. . 3 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒))
27	1, 2, 26	mp2and 712	. 2 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒)
28	27	3expib 1140	1 ⊢ (𝐺 ∈ ComplGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∖ cdif 4048  {csn 4588  {cpr 4590  ‘cfv 6566  Vtxcvtx 29631  Edgcedg 29682  ComplGraphccplgr 30047

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-nbgr 29971  df-uvtx 30024  df-cplgr 30049

This theorem is used by:  cusgredgex  36104