![]() |
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 4507 | . . 3 β’ ((VtxβπΊ) = β β βπ β (VtxβπΊ)βπ β (VtxβπΊ)βπβπ π(π(PathsOnβπΊ)π)π) | |
2 | 1 | adantl 480 | . 2 β’ ((πΊ β π β§ (VtxβπΊ) = β ) β βπ β (VtxβπΊ)βπ β (VtxβπΊ)βπβπ π(π(PathsOnβπΊ)π)π) |
3 | eqid 2730 | . . . 4 β’ (VtxβπΊ) = (VtxβπΊ) | |
4 | 3 | isconngr 29709 | . . 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 479 | . 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 256 | 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 394 = wceq 1539 βwex 1779 β wcel 2104 βwral 3059 β c0 4321 class class class wbr 5147 βcfv 6542 (class class class)co 7411 Vtxcvtx 28523 PathsOncpthson 29238 ConnGraphcconngr 29706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-nul 5305 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6494 df-fv 6550 df-ov 7414 df-conngr 29707 |
This theorem is referenced by: 1conngr 29714 |
Copyright terms: Public domain | W3C validator |