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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 27624, 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 5383 | . . 3 ⊢ 〈𝑉, ∅〉 ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ∅〉 ∈ V) |
3 | 0ex 5235 | . . 3 ⊢ ∅ ∈ V | |
4 | opiedgfv 27388 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ ∅ ∈ V) → (iEdg‘〈𝑉, ∅〉) = ∅) | |
5 | 3, 4 | mpan2 688 | . 2 ⊢ (𝑉 ∈ 𝑊 → (iEdg‘〈𝑉, ∅〉) = ∅) |
6 | 2, 5 | upgr0e 27492 | 1 ⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ∅〉 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ∅c0 4262 〈cop 4573 ‘cfv 6432 iEdgciedg 27378 UPGraphcupgr 27461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-i2m1 10950 ax-1ne0 10951 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7275 df-2nd 7826 df-er 8490 df-en 8726 df-dom 8727 df-sdom 8728 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-2 12047 df-iedg 27380 df-upgr 27463 df-umgr 27464 |
This theorem is referenced by: (None) |
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