Theorem uvtxnbgr 29380

Description: A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 23-Mar-2021.)

Hypothesis

Ref	Expression
uvtxnbgr.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
uvtxnbgr	⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))

Proof of Theorem uvtxnbgr

Step	Hyp	Ref	Expression
1		uvtxnbgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	nbgrssovtx 29341	. . 3 ⊢ (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁})
3	2	a1i 11	. 2 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}))
4	1	uvtxnbgrss 29372	. 2 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝑉 ∖ {𝑁}) ⊆ (𝐺 NeighbVtx 𝑁))
5	3, 4	eqssd 3948	1 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   ∖ cdif 3895   ⊆ wss 3898  {csn 4575  ‘cfv 6486  (class class class)co 7352  Vtxcvtx 28976   NeighbVtx cnbgr 29312  UnivVtxcuvtx 29365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-nbgr 29313  df-uvtx 29366

This theorem is referenced by:  uvtxnbgrb  29381  uvtxnm1nbgr  29384  uvtxupgrres  29388