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| Mirrors > Home > MPE Home > Th. List > upgr0e | Structured version Visualization version GIF version | ||
| Description: The empty graph, with vertices but no edges, is a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 11-Oct-2020.) (Proof shortened by AV, 25-Nov-2020.) | 
| Ref | Expression | 
|---|---|
| umgr0e.g | ⊢ (𝜑 → 𝐺 ∈ 𝑊) | 
| umgr0e.e | ⊢ (𝜑 → (iEdg‘𝐺) = ∅) | 
| Ref | Expression | 
|---|---|
| upgr0e | ⊢ (𝜑 → 𝐺 ∈ UPGraph) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | umgr0e.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
| 2 | umgr0e.e | . . 3 ⊢ (𝜑 → (iEdg‘𝐺) = ∅) | |
| 3 | 1, 2 | umgr0e 29128 | . 2 ⊢ (𝜑 → 𝐺 ∈ UMGraph) | 
| 4 | umgrupgr 29121 | . 2 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph) | |
| 5 | 3, 4 | syl 17 | 1 ⊢ (𝜑 → 𝐺 ∈ UPGraph) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∅c0 4332 ‘cfv 6560 iEdgciedg 29015 UPGraphcupgr 29098 UMGraphcumgr 29099 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-i2m1 11224 ax-1ne0 11225 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-2 12330 df-upgr 29100 df-umgr 29101 | 
| This theorem is referenced by: upgr0eop 29132 upgr0eopALT 29134 | 
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