Theorem uhgr0v0e 29321

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 2742	. . . . 5 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅)
3		uhgr0vb 29155	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅))
4	3	biimpd 229	. . . . . 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 242	. . . 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 2742	. . . 4 ⊢ (𝐸 = ∅ ↔ (Edg‘𝐺) = ∅)
11		uhgriedg0edg0 29210	. . . 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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∅c0 4274  ‘cfv 6492  Vtxcvtx 29079  iEdgciedg 29080  Edgcedg 29130  UHGraphcuhgr 29139

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 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-edg 29131  df-uhgr 29141

This theorem is referenced by:  uhgr0vsize0  29322  uhgr0vusgr  29325  fusgrfisbase  29411