Theorem prcliscplgr 27198

Description: A proper class (representing a null graph, see vtxvalprc 26832) has the property of a complete graph (see also cplgr0v 27211), but cannot be an element of ComplGraph, of course. Because of this, a sethood antecedent like 𝐺 ∈ 𝑊 is necessary in the following theorems like iscplgr 27199. (Contributed by AV, 14-Feb-2022.)

Hypothesis

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

Assertion

Ref	Expression
prcliscplgr	⊢ (¬ 𝐺 ∈ V → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))

Distinct variable groups:   𝑣,𝐺   𝑣,𝑉

Proof of Theorem prcliscplgr

Step	Hyp	Ref	Expression
1		fvprc 6665	. 2 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = ∅)
2		cplgruvtxb.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	eqeq1i 2828	. . 3 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅)
4		rzal 4455	. . 3 ⊢ (𝑉 = ∅ → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
5	3, 4	sylbir 237	. 2 ⊢ ((Vtx‘𝐺) = ∅ → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
6	1, 5	syl 17	1 ⊢ (¬ 𝐺 ∈ V → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   ∈ wcel 2114  ∀wral 3140  Vcvv 3496  ∅c0 4293  ‘cfv 6357  Vtxcvtx 26783  UnivVtxcuvtx 27169

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-nul 5212  ax-pow 5268

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365

This theorem is referenced by: (None)