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Theorem cplgredgex 32546
 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 → ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {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.
StepHypRef Expression
1 simp2 1134 . . 3 ((𝐺 ∈ ComplGraph ∧ 𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐴𝑉)
2 simp3 1135 . . 3 ((𝐺 ∈ ComplGraph ∧ 𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐵 ∈ (𝑉 ∖ {𝐴}))
3 eleq1 2877 . . . . . . 7 (𝑎 = 𝐴 → (𝑎𝑉𝐴𝑉))
4 sneq 4538 . . . . . . . . 9 (𝑎 = 𝐴 → {𝑎} = {𝐴})
54difeq2d 4053 . . . . . . . 8 (𝑎 = 𝐴 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝐴}))
65eleq2d 2875 . . . . . . 7 (𝑎 = 𝐴 → (𝑏 ∈ (𝑉 ∖ {𝑎}) ↔ 𝑏 ∈ (𝑉 ∖ {𝐴})))
73, 6anbi12d 633 . . . . . 6 (𝑎 = 𝐴 → ((𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})) ↔ (𝐴𝑉𝑏 ∈ (𝑉 ∖ {𝐴}))))
8 preq1 4632 . . . . . . . 8 (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏})
98sseq1d 3948 . . . . . . 7 (𝑎 = 𝐴 → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝐴, 𝑏} ⊆ 𝑒))
109rexbidv 3257 . . . . . 6 (𝑎 = 𝐴 → (∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝐴, 𝑏} ⊆ 𝑒))
117, 10imbi12d 348 . . . . 5 (𝑎 = 𝐴 → (((𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})) → ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒) ↔ ((𝐴𝑉𝑏 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝑏} ⊆ 𝑒)))
12 eleq1 2877 . . . . . . 7 (𝑏 = 𝐵 → (𝑏 ∈ (𝑉 ∖ {𝐴}) ↔ 𝐵 ∈ (𝑉 ∖ {𝐴})))
1312anbi2d 631 . . . . . 6 (𝑏 = 𝐵 → ((𝐴𝑉𝑏 ∈ (𝑉 ∖ {𝐴})) ↔ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴}))))
14 preq2 4633 . . . . . . . 8 (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵})
1514sseq1d 3948 . . . . . . 7 (𝑏 = 𝐵 → ({𝐴, 𝑏} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ 𝑒))
1615rexbidv 3257 . . . . . 6 (𝑏 = 𝐵 → (∃𝑒𝐸 {𝐴, 𝑏} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒))
1713, 16imbi12d 348 . . . . 5 (𝑏 = 𝐵 → (((𝐴𝑉𝑏 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝑏} ⊆ 𝑒) ↔ ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒)))
1811, 17sylan9bb 513 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → (((𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})) → ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒) ↔ ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒)))
19 cplgredgex.1 . . . . . . . 8 𝑉 = (Vtx‘𝐺)
20 cplgredgex.2 . . . . . . . 8 𝐸 = (Edg‘𝐺)
2119, 20iscplgredg 27251 . . . . . . 7 (𝐺 ∈ ComplGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒))
2221ibi 270 . . . . . 6 (𝐺 ∈ ComplGraph → ∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒)
23 rsp2 3177 . . . . . 6 (∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒 → ((𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})) → ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒))
2422, 23syl 17 . . . . 5 (𝐺 ∈ ComplGraph → ((𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})) → ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒))
25243ad2ant1 1130 . . . 4 ((𝐺 ∈ ComplGraph ∧ 𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ((𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})) → ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒))
261, 2, 18, 25vtocl2d 3506 . . 3 ((𝐺 ∈ ComplGraph ∧ 𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒))
271, 2, 26mp2and 698 . 2 ((𝐺 ∈ ComplGraph ∧ 𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒)
28273expib 1119 1 (𝐺 ∈ ComplGraph → ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ∖ cdif 3880   ⊆ wss 3883  {csn 4528  {cpr 4530  ‘cfv 6332  Vtxcvtx 26833  Edgcedg 26884  ComplGraphccplgr 27243 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7684  df-2nd 7685  df-nbgr 27167  df-uvtx 27220  df-cplgr 27245 This theorem is referenced by:  cusgredgex  32547
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