uhgr0v0e - Metamath Proof Explorer

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Theorem uhgr0v0e 29255

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 29089	. . . . . . 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 29144	. . . 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 = ∅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∅c0 4333 ‘cfv 6561 Vtxcvtx 29013 iEdgciedg 29014 Edgcedg 29064 UHGraphcuhgr 29073

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-edg 29065 df-uhgr 29075

This theorem is referenced by: uhgr0vsize0 29256 uhgr0vusgr 29259 fusgrfisbase 29345