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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 4503 | . . 3 β’ ((VtxβπΊ) = β β βπ β (VtxβπΊ)βπ β (VtxβπΊ)βπβπ π(π(PathsOnβπΊ)π)π) | |
2 | 1 | adantl 481 | . 2 β’ ((πΊ β π β§ (VtxβπΊ) = β ) β βπ β (VtxβπΊ)βπ β (VtxβπΊ)βπβπ π(π(PathsOnβπΊ)π)π) |
3 | eqid 2726 | . . . 4 β’ (VtxβπΊ) = (VtxβπΊ) | |
4 | 3 | isconngr 29951 | . . 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 480 | . 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 395 = wceq 1533 βwex 1773 β wcel 2098 βwral 3055 β c0 4317 class class class wbr 5141 βcfv 6537 (class class class)co 7405 Vtxcvtx 28764 PathsOncpthson 29480 ConnGraphcconngr 29948 |
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 2697 ax-nul 5299 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-iota 6489 df-fv 6545 df-ov 7408 df-conngr 29949 |
This theorem is referenced by: 1conngr 29956 |
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