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| Mirrors > Home > MPE Home > Th. List > vtxdg0v | Structured version Visualization version GIF version | ||
| Description: The degree of a vertex in the null graph is zero (or anything else), because there are no vertices. (Contributed by AV, 11-Dec-2020.) |
| Ref | Expression |
|---|---|
| vtxdgf.v | β’ π = (VtxβπΊ) |
| Ref | Expression |
|---|---|
| vtxdg0v | β’ ((πΊ = β β§ π β π) β ((VtxDegβπΊ)βπ) = 0) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vtxdgf.v | . . . . 5 β’ π = (VtxβπΊ) | |
| 2 | 1 | eleq2i 2824 | . . . 4 β’ (π β π β π β (VtxβπΊ)) |
| 3 | fveq2 6891 | . . . . . 6 β’ (πΊ = β β (VtxβπΊ) = (Vtxββ )) | |
| 4 | vtxval0 28734 | . . . . . 6 β’ (Vtxββ ) = β | |
| 5 | 3, 4 | eqtrdi 2787 | . . . . 5 β’ (πΊ = β β (VtxβπΊ) = β ) |
| 6 | 5 | eleq2d 2818 | . . . 4 β’ (πΊ = β β (π β (VtxβπΊ) β π β β )) |
| 7 | 2, 6 | bitrid 283 | . . 3 β’ (πΊ = β β (π β π β π β β )) |
| 8 | noel 4330 | . . . 4 β’ Β¬ π β β | |
| 9 | 8 | pm2.21i 119 | . . 3 β’ (π β β β ((VtxDegβπΊ)βπ) = 0) |
| 10 | 7, 9 | syl6bi 253 | . 2 β’ (πΊ = β β (π β π β ((VtxDegβπΊ)βπ) = 0)) |
| 11 | 10 | imp 406 | 1 β’ ((πΊ = β β§ π β π) β ((VtxDegβπΊ)βπ) = 0) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 β c0 4322 βcfv 6543 0cc0 11116 Vtxcvtx 28691 VtxDegcvtxdg 29157 |
| 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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-1cn 11174 ax-addcl 11176 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-nn 12220 df-slot 17122 df-ndx 17134 df-base 17152 df-vtx 28693 |
| This theorem is referenced by: (None) |
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