Theorem uhgr0v0e 29003

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 2731	. . . . 5 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅)
3		uhgr0vb 28840	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅))
4	3	biimpd 228	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅))
5	4	ex 412	. . . . 5 ⊢ (𝐺 ∈ UHGraph → ((Vtx‘𝐺) = ∅ → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅)))
6	2, 5	biimtrid 241	. . . 4 ⊢ (𝐺 ∈ UHGraph → (𝑉 = ∅ → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅)))
7	6	pm2.43a 54	. . 3 ⊢ (𝐺 ∈ UHGraph → (𝑉 = ∅ → (iEdg‘𝐺) = ∅))
8	7	imp 406	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (iEdg‘𝐺) = ∅)
9		uhgr0v0e.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
10	9	eqeq1i 2731	. . . 4 ⊢ (𝐸 = ∅ ↔ (Edg‘𝐺) = ∅)
11		uhgriedg0edg0 28895	. . . 4 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
12	10, 11	bitrid 283	. . 3 ⊢ (𝐺 ∈ UHGraph → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅))
13	12	adantr 480	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅))
14	8, 13	mpbird 257	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → 𝐸 = ∅)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∅c0 4317  ‘cfv 6537  Vtxcvtx 28764  iEdgciedg 28765  Edgcedg 28815  UHGraphcuhgr 28824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-edg 28816  df-uhgr 28826

This theorem is referenced by:  uhgr0vsize0  29004  uhgr0vusgr  29007  fusgrfisbase  29093