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Theorem cplgredgex 35105
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 1136 . . 3 ((𝐺 ∈ ComplGraph ∧ 𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐴𝑉)
2 simp3 1137 . . 3 ((𝐺 ∈ ComplGraph ∧ 𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐵 ∈ (𝑉 ∖ {𝐴}))
3 eleq1 2827 . . . . . . 7 (𝑎 = 𝐴 → (𝑎𝑉𝐴𝑉))
4 sneq 4641 . . . . . . . . 9 (𝑎 = 𝐴 → {𝑎} = {𝐴})
54difeq2d 4136 . . . . . . . 8 (𝑎 = 𝐴 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝐴}))
65eleq2d 2825 . . . . . . 7 (𝑎 = 𝐴 → (𝑏 ∈ (𝑉 ∖ {𝑎}) ↔ 𝑏 ∈ (𝑉 ∖ {𝐴})))
73, 6anbi12d 632 . . . . . 6 (𝑎 = 𝐴 → ((𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})) ↔ (𝐴𝑉𝑏 ∈ (𝑉 ∖ {𝐴}))))
8 preq1 4738 . . . . . . . 8 (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏})
98sseq1d 4027 . . . . . . 7 (𝑎 = 𝐴 → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝐴, 𝑏} ⊆ 𝑒))
109rexbidv 3177 . . . . . 6 (𝑎 = 𝐴 → (∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝐴, 𝑏} ⊆ 𝑒))
117, 10imbi12d 344 . . . . 5 (𝑎 = 𝐴 → (((𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})) → ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒) ↔ ((𝐴𝑉𝑏 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝑏} ⊆ 𝑒)))
12 eleq1 2827 . . . . . . 7 (𝑏 = 𝐵 → (𝑏 ∈ (𝑉 ∖ {𝐴}) ↔ 𝐵 ∈ (𝑉 ∖ {𝐴})))
1312anbi2d 630 . . . . . 6 (𝑏 = 𝐵 → ((𝐴𝑉𝑏 ∈ (𝑉 ∖ {𝐴})) ↔ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴}))))
14 preq2 4739 . . . . . . . 8 (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵})
1514sseq1d 4027 . . . . . . 7 (𝑏 = 𝐵 → ({𝐴, 𝑏} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ 𝑒))
1615rexbidv 3177 . . . . . 6 (𝑏 = 𝐵 → (∃𝑒𝐸 {𝐴, 𝑏} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒))
1713, 16imbi12d 344 . . . . 5 (𝑏 = 𝐵 → (((𝐴𝑉𝑏 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝑏} ⊆ 𝑒) ↔ ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒)))
1811, 17sylan9bb 509 . . . 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 29449 . . . . . . 7 (𝐺 ∈ ComplGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒))
2221ibi 267 . . . . . 6 (𝐺 ∈ ComplGraph → ∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒)
23 rsp2 3275 . . . . . 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 1132 . . . 4 ((𝐺 ∈ ComplGraph ∧ 𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ((𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})) → ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒))
261, 2, 18, 25vtocl2d 3562 . . 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 1121 1 (𝐺 ∈ ComplGraph → ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  cdif 3960  wss 3963  {csn 4631  {cpr 4633  cfv 6563  Vtxcvtx 29028  Edgcedg 29079  ComplGraphccplgr 29441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-nbgr 29365  df-uvtx 29418  df-cplgr 29443
This theorem is referenced by:  cusgredgex  35106
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