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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 4512 | . . 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 2728 | . . . 4 β’ (VtxβπΊ) = (VtxβπΊ) | |
4 | 3 | isconngr 30027 | . . 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 1533 βwex 1773 β wcel 2098 βwral 3058 β c0 4326 class class class wbr 5152 βcfv 6553 (class class class)co 7426 Vtxcvtx 28837 PathsOncpthson 29556 ConnGraphcconngr 30024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-nul 5310 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-iota 6505 df-fv 6561 df-ov 7429 df-conngr 30025 |
This theorem is referenced by: 1conngr 30032 |
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