Theorem uvtxel1 29465

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 4577	. . . 4 ⊢ (𝑛 = 𝑁 → {𝑛} = {𝑁})
2	1	difeq2d 4066	. . 3 ⊢ (𝑛 = 𝑁 → (𝑉 ∖ {𝑛}) = (𝑉 ∖ {𝑁}))
3		preq2 4678	. . . . 5 ⊢ (𝑛 = 𝑁 → {𝑘, 𝑛} = {𝑘, 𝑁})
4	3	sseq1d 3953	. . . 4 ⊢ (𝑛 = 𝑁 → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑘, 𝑁} ⊆ 𝑒))
5	4	rexbidv 3161	. . 3 ⊢ (𝑛 = 𝑁 → (∃𝑒 ∈ 𝐸 {𝑘, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑘, 𝑁} ⊆ 𝑒))
6	2, 5	raleqbidv 3311	. 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 29464	. 2 ⊢ (UnivVtx‘𝐺) = {𝑛 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛})∃𝑒 ∈ 𝐸 {𝑘, 𝑛} ⊆ 𝑒}
10	6, 9	elrab2 3637	1 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒 ∈ 𝐸 {𝑘, 𝑁} ⊆ 𝑒))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061   ∖ cdif 3886   ⊆ wss 3889  {csn 4567  {cpr 4569  ‘cfv 6498  Vtxcvtx 29065  Edgcedg 29116  UnivVtxcuvtx 29454

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-nbgr 29402  df-uvtx 29455

This theorem is referenced by: (None)