Theorem uhgr0v0e 29497

Description: The null graph, with no vertices, has no edges. (Contributed by AV, 21-Oct-2020.)

Hypotheses

Ref	Expression
uhgr0v0e.v	⊢ 𝑉 = (Vtx‘𝐺)
uhgr0v0e.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
uhgr0v0e	⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → 𝐸 = ∅)

Proof of Theorem uhgr0v0e

Step	Hyp	Ref	Expression
1		uhgr0v0e.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	eqeq1i 2770	. . . . 5 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅)
3		uhgr0vb 29331	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅))
4	3	biimpd 232	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅))
5	4	ex 417	. . . . 5 ⊢ (𝐺 ∈ UHGraph → ((Vtx‘𝐺) = ∅ → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅)))
6	2, 5	biimtrid 245	. . . 4 ⊢ (𝐺 ∈ UHGraph → (𝑉 = ∅ → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅)))
7	6	pm2.43a 55	. . 3 ⊢ (𝐺 ∈ UHGraph → (𝑉 = ∅ → (iEdg‘𝐺) = ∅))
8	7	imp 411	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (iEdg‘𝐺) = ∅)
9		uhgr0v0e.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
10	9	eqeq1i 2770	. . . 4 ⊢ (𝐸 = ∅ ↔ (Edg‘𝐺) = ∅)
11		uhgriedg0edg0 29386	. . . 4 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
12	10, 11	bitrid 286	. . 3 ⊢ (𝐺 ∈ UHGraph → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅))
13	12	adantr 485	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅))
14	8, 13	mpbird 260	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → 𝐸 = ∅)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∅c0 4288  ‘cfv 6525  Vtxcvtx 29255  iEdgciedg 29256  Edgcedg 29306  UHGraphcuhgr 29315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533  df-edg 29307  df-uhgr 29317

This theorem is referenced by:  uhgr0vsize0  29498  uhgr0vusgr  29501  fusgrfisbase  29587