![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 0vconngr | Structured version Visualization version GIF version |
Description: A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
Ref | Expression |
---|---|
0vconngr | β’ ((πΊ β π β§ (VtxβπΊ) = β ) β πΊ β ConnGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rzal 4467 | . . 3 β’ ((VtxβπΊ) = β β βπ β (VtxβπΊ)βπ β (VtxβπΊ)βπβπ π(π(PathsOnβπΊ)π)π) | |
2 | 1 | adantl 483 | . 2 β’ ((πΊ β π β§ (VtxβπΊ) = β ) β βπ β (VtxβπΊ)βπ β (VtxβπΊ)βπβπ π(π(PathsOnβπΊ)π)π) |
3 | eqid 2737 | . . . 4 β’ (VtxβπΊ) = (VtxβπΊ) | |
4 | 3 | isconngr 29136 | . . 3 β’ (πΊ β π β (πΊ β ConnGraph β βπ β (VtxβπΊ)βπ β (VtxβπΊ)βπβπ π(π(PathsOnβπΊ)π)π)) |
5 | 4 | adantr 482 | . 2 β’ ((πΊ β π β§ (VtxβπΊ) = β ) β (πΊ β ConnGraph β βπ β (VtxβπΊ)βπ β (VtxβπΊ)βπβπ π(π(PathsOnβπΊ)π)π)) |
6 | 2, 5 | mpbird 257 | 1 β’ ((πΊ β π β§ (VtxβπΊ) = β ) β πΊ β ConnGraph) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 βwex 1782 β wcel 2107 βwral 3065 β c0 4283 class class class wbr 5106 βcfv 6497 (class class class)co 7358 Vtxcvtx 27950 PathsOncpthson 28665 ConnGraphcconngr 29133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-nul 5264 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rab 3409 df-v 3448 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 df-conngr 29134 |
This theorem is referenced by: 1conngr 29141 |
Copyright terms: Public domain | W3C validator |