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Mirrors > Home > MPE Home > Th. List > upgr0eop | Structured version Visualization version GIF version |
Description: The empty graph, with vertices but no edges, is a pseudograph. The empty graph is actually a simple graph, see usgr0eop 27014, and therefore also a multigraph (𝐺 ∈ UMGraph). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 11-Oct-2020.) |
Ref | Expression |
---|---|
upgr0eop | ⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ∅〉 ∈ UPGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5342 | . . 3 ⊢ 〈𝑉, ∅〉 ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ∅〉 ∈ V) |
3 | 0ex 5197 | . . 3 ⊢ ∅ ∈ V | |
4 | opiedgfv 26778 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ ∅ ∈ V) → (iEdg‘〈𝑉, ∅〉) = ∅) | |
5 | 3, 4 | mpan2 689 | . 2 ⊢ (𝑉 ∈ 𝑊 → (iEdg‘〈𝑉, ∅〉) = ∅) |
6 | 2, 5 | upgr0e 26882 | 1 ⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ∅〉 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3486 ∅c0 4279 〈cop 4559 ‘cfv 6341 iEdgciedg 26768 UPGraphcupgr 26851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-i2m1 10591 ax-1ne0 10592 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-ov 7145 df-2nd 7676 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-2 11687 df-iedg 26770 df-upgr 26853 df-umgr 26854 |
This theorem is referenced by: (None) |
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