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Theorem uvtxel1 26520
 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 4331 . . . 4 (𝑛 = 𝑁 → {𝑛} = {𝑁})
21difeq2d 3871 . . 3 (𝑛 = 𝑁 → (𝑉 ∖ {𝑛}) = (𝑉 ∖ {𝑁}))
3 preq2 4413 . . . . 5 (𝑛 = 𝑁 → {𝑘, 𝑛} = {𝑘, 𝑁})
43sseq1d 3773 . . . 4 (𝑛 = 𝑁 → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑘, 𝑁} ⊆ 𝑒))
54rexbidv 3190 . . 3 (𝑛 = 𝑁 → (∃𝑒𝐸 {𝑘, 𝑛} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝑘, 𝑁} ⊆ 𝑒))
62, 5raleqbidv 3291 . 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 26518 . 2 (UnivVtx‘𝐺) = {𝑛𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛})∃𝑒𝐸 {𝑘, 𝑛} ⊆ 𝑒}
106, 9elrab2 3507 1 (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒𝐸 {𝑘, 𝑁} ⊆ 𝑒))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051   ∖ cdif 3712   ⊆ wss 3715  {csn 4321  {cpr 4323  ‘cfv 6049  Vtxcvtx 26094  Edgcedg 26159  UnivVtxcuvtx 26506 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-nbgr 26445  df-uvtx 26507 This theorem is referenced by: (None)
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