Theorem uvtxel1 29686

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) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒 ∈ 𝐸 {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.

Step	Hyp	Ref	Expression
1		sneq 4604	. . . 4 ⊢ (𝑛 = 𝑁 → {𝑛} = {𝑁})
2	1	difeq2d 4089	. . 3 ⊢ (𝑛 = 𝑁 → (𝑉 ∖ {𝑛}) = (𝑉 ∖ {𝑁}))
3		preq2 4705	. . . . 5 ⊢ (𝑛 = 𝑁 → {𝑘, 𝑛} = {𝑘, 𝑁})
4	3	sseq1d 3976	. . . 4 ⊢ (𝑛 = 𝑁 → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑘, 𝑁} ⊆ 𝑒))
5	4	rexbidv 3195	. . 3 ⊢ (𝑛 = 𝑁 → (∃𝑒 ∈ 𝐸 {𝑘, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑘, 𝑁} ⊆ 𝑒))
6	2, 5	raleqbidv 3345	. 2 ⊢ (𝑛 = 𝑁 → (∀𝑘 ∈ (𝑉 ∖ {𝑛})∃𝑒 ∈ 𝐸 {𝑘, 𝑛} ⊆ 𝑒 ↔ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒 ∈ 𝐸 {𝑘, 𝑁} ⊆ 𝑒))
7		uvtxel.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
8		isuvtx.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
9	7, 8	isuvtx 29685	. 2 ⊢ (UnivVtx‘𝐺) = {𝑛 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛})∃𝑒 ∈ 𝐸 {𝑘, 𝑛} ⊆ 𝑒}
10	6, 9	elrab2 3663	1 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒 ∈ 𝐸 {𝑘, 𝑁} ⊆ 𝑒))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095   ∖ cdif 3910   ⊆ wss 3913  {csn 4594  {cpr 4596  ‘cfv 6537  Vtxcvtx 29286  Edgcedg 29337  UnivVtxcuvtx 29675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986  df-nbgr 29623  df-uvtx 29676

This theorem is referenced by: (None)