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Mirrors > Home > MPE Home > Th. List > vtxval0 | Structured version Visualization version GIF version |
Description: Degenerated case 1 for vertices: The set of vertices of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.) |
Ref | Expression |
---|---|
vtxval0 | ⊢ (Vtx‘∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nelxp 5389 | . . 3 ⊢ ¬ ∅ ∈ (V × V) | |
2 | 1 | iffalsei 4317 | . 2 ⊢ if(∅ ∈ (V × V), (1st ‘∅), (Base‘∅)) = (Base‘∅) |
3 | vtxval 26348 | . 2 ⊢ (Vtx‘∅) = if(∅ ∈ (V × V), (1st ‘∅), (Base‘∅)) | |
4 | base0 16308 | . 2 ⊢ ∅ = (Base‘∅) | |
5 | 2, 3, 4 | 3eqtr4i 2812 | 1 ⊢ (Vtx‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2107 Vcvv 3398 ∅c0 4141 ifcif 4307 × cxp 5353 ‘cfv 6135 1st c1st 7443 Basecbs 16255 Vtxcvtx 26344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-slot 16259 df-base 16261 df-vtx 26346 |
This theorem is referenced by: uhgr0 26421 usgr0 26590 0grsubgr 26625 cplgr0 26773 vtxdg0v 26821 0grrusgr 26927 0wlk0 27000 0conngr 27595 frgr0 27672 |
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