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Theorem cplgredgex 32795
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 1139 . . 3 ((𝐺 ∈ ComplGraph ∧ 𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐴𝑉)
2 simp3 1140 . . 3 ((𝐺 ∈ ComplGraph ∧ 𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐵 ∈ (𝑉 ∖ {𝐴}))
3 eleq1 2825 . . . . . . 7 (𝑎 = 𝐴 → (𝑎𝑉𝐴𝑉))
4 sneq 4551 . . . . . . . . 9 (𝑎 = 𝐴 → {𝑎} = {𝐴})
54difeq2d 4037 . . . . . . . 8 (𝑎 = 𝐴 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝐴}))
65eleq2d 2823 . . . . . . 7 (𝑎 = 𝐴 → (𝑏 ∈ (𝑉 ∖ {𝑎}) ↔ 𝑏 ∈ (𝑉 ∖ {𝐴})))
73, 6anbi12d 634 . . . . . 6 (𝑎 = 𝐴 → ((𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})) ↔ (𝐴𝑉𝑏 ∈ (𝑉 ∖ {𝐴}))))
8 preq1 4649 . . . . . . . 8 (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏})
98sseq1d 3932 . . . . . . 7 (𝑎 = 𝐴 → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝐴, 𝑏} ⊆ 𝑒))
109rexbidv 3216 . . . . . 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 2825 . . . . . . 7 (𝑏 = 𝐵 → (𝑏 ∈ (𝑉 ∖ {𝐴}) ↔ 𝐵 ∈ (𝑉 ∖ {𝐴})))
1312anbi2d 632 . . . . . 6 (𝑏 = 𝐵 → ((𝐴𝑉𝑏 ∈ (𝑉 ∖ {𝐴})) ↔ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴}))))
14 preq2 4650 . . . . . . . 8 (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵})
1514sseq1d 3932 . . . . . . 7 (𝑏 = 𝐵 → ({𝐴, 𝑏} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ 𝑒))
1615rexbidv 3216 . . . . . 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 27505 . . . . . . 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 3134 . . . . . 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 1135 . . . 4 ((𝐺 ∈ ComplGraph ∧ 𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ((𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})) → ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒))
261, 2, 18, 25vtocl2d 3472 . . 3 ((𝐺 ∈ ComplGraph ∧ 𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒))
271, 2, 26mp2and 699 . 2 ((𝐺 ∈ ComplGraph ∧ 𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒)
28273expib 1124 1 (𝐺 ∈ ComplGraph → ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  wrex 3062  cdif 3863  wss 3866  {csn 4541  {cpr 4543  cfv 6380  Vtxcvtx 27087  Edgcedg 27138  ComplGraphccplgr 27497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-nbgr 27421  df-uvtx 27474  df-cplgr 27499
This theorem is referenced by:  cusgredgex  32796
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