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Theorem uvtxel1 27184
Description: Characterization of a universal vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 14-Feb-2022.)
Hypotheses
Ref Expression
uvtxel.v 𝑉 = (Vtx‘𝐺)
isuvtx.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
uvtxel1 (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒𝐸 {𝑘, 𝑁} ⊆ 𝑒))
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺,𝑘   𝑒,𝑉,𝑘   𝑒,𝑁,𝑘
Allowed substitution hint:   𝐸(𝑘)

Proof of Theorem uvtxel1
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 sneq 4549 . . . 4 (𝑛 = 𝑁 → {𝑛} = {𝑁})
21difeq2d 4074 . . 3 (𝑛 = 𝑁 → (𝑉 ∖ {𝑛}) = (𝑉 ∖ {𝑁}))
3 preq2 4644 . . . . 5 (𝑛 = 𝑁 → {𝑘, 𝑛} = {𝑘, 𝑁})
43sseq1d 3973 . . . 4 (𝑛 = 𝑁 → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑘, 𝑁} ⊆ 𝑒))
54rexbidv 3283 . . 3 (𝑛 = 𝑁 → (∃𝑒𝐸 {𝑘, 𝑛} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝑘, 𝑁} ⊆ 𝑒))
62, 5raleqbidv 3382 . 2 (𝑛 = 𝑁 → (∀𝑘 ∈ (𝑉 ∖ {𝑛})∃𝑒𝐸 {𝑘, 𝑛} ⊆ 𝑒 ↔ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒𝐸 {𝑘, 𝑁} ⊆ 𝑒))
7 uvtxel.v . . 3 𝑉 = (Vtx‘𝐺)
8 isuvtx.e . . 3 𝐸 = (Edg‘𝐺)
97, 8isuvtx 27183 . 2 (UnivVtx‘𝐺) = {𝑛𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛})∃𝑒𝐸 {𝑘, 𝑛} ⊆ 𝑒}
106, 9elrab2 3658 1 (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒𝐸 {𝑘, 𝑁} ⊆ 𝑒))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2114  wral 3130  wrex 3131  cdif 3905  wss 3908  {csn 4539  {cpr 4541  cfv 6334  Vtxcvtx 26787  Edgcedg 26838  UnivVtxcuvtx 27173
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-nbgr 27121  df-uvtx 27174
This theorem is referenced by: (None)
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