uhgr0v0e - Metamath Proof Explorer

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Theorem uhgr0v0e 29801

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 2766	. . . . 5 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅)
3		uhgr0vb 29632	. . . . . . 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 418	. . . . 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 412	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (iEdg‘𝐺) = ∅)
9		uhgr0v0e.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
10	9	eqeq1i 2766	. . . 4 ⊢ (𝐸 = ∅ ↔ (Edg‘𝐺) = ∅)
11		uhgriedg0edg0 29687	. . . 4 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
12	10, 11	bitrid 286	. . 3 ⊢ (𝐺 ∈ UHGraph → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅))
13	12	adantr 486	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅))
14	8, 13	mpbird 260	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → 𝐸 = ∅)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∅c0 4279 ‘cfv 6531 Vtxcvtx 29556 iEdgciedg 29557 Edgcedg 29607 UHGraphcuhgr 29616

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pr 5391 ax-un 7740

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-sbc 3740 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-fv 6539 df-edg 29608 df-uhgr 29618

This theorem is used by: uhgr0vsize0 29802 uhgr0vusgr 29805 fusgrfisbase 29891

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