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Theorem cplgredgex 34111
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 1138 . . 3 ((𝐺 ∈ ComplGraph ∧ 𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐴𝑉)
2 simp3 1139 . . 3 ((𝐺 ∈ ComplGraph ∧ 𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐵 ∈ (𝑉 ∖ {𝐴}))
3 eleq1 2822 . . . . . . 7 (𝑎 = 𝐴 → (𝑎𝑉𝐴𝑉))
4 sneq 4639 . . . . . . . . 9 (𝑎 = 𝐴 → {𝑎} = {𝐴})
54difeq2d 4123 . . . . . . . 8 (𝑎 = 𝐴 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝐴}))
65eleq2d 2820 . . . . . . 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 4014 . . . . . . 7 (𝑎 = 𝐴 → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝐴, 𝑏} ⊆ 𝑒))
109rexbidv 3179 . . . . . 6 (𝑎 = 𝐴 → (∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝐴, 𝑏} ⊆ 𝑒))
117, 10imbi12d 345 . . . . 5 (𝑎 = 𝐴 → (((𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})) → ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒) ↔ ((𝐴𝑉𝑏 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝑏} ⊆ 𝑒)))
12 eleq1 2822 . . . . . . 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 4014 . . . . . . 7 (𝑏 = 𝐵 → ({𝐴, 𝑏} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ 𝑒))
1615rexbidv 3179 . . . . . 6 (𝑏 = 𝐵 → (∃𝑒𝐸 {𝐴, 𝑏} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒))
1713, 16imbi12d 345 . . . . 5 (𝑏 = 𝐵 → (((𝐴𝑉𝑏 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝑏} ⊆ 𝑒) ↔ ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒)))
1811, 17sylan9bb 511 . . . 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 28674 . . . . . . 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 1134 . . . 4 ((𝐺 ∈ ComplGraph ∧ 𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ((𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})) → ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒))
261, 2, 18, 25vtocl2d 3545 . . 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 1123 1 (𝐺 ∈ ComplGraph → ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  wrex 3071  cdif 3946  wss 3949  {csn 4629  {cpr 4631  cfv 6544  Vtxcvtx 28256  Edgcedg 28307  ComplGraphccplgr 28666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-nbgr 28590  df-uvtx 28643  df-cplgr 28668
This theorem is referenced by:  cusgredgex  34112
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