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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 4508 | . . 3 β’ ((VtxβπΊ) = β β βπ β (VtxβπΊ)βπ β (VtxβπΊ)βπβπ π(π(PathsOnβπΊ)π)π) | |
2 | 1 | adantl 482 | . 2 β’ ((πΊ β π β§ (VtxβπΊ) = β ) β βπ β (VtxβπΊ)βπ β (VtxβπΊ)βπβπ π(π(PathsOnβπΊ)π)π) |
3 | eqid 2732 | . . . 4 β’ (VtxβπΊ) = (VtxβπΊ) | |
4 | 3 | isconngr 29439 | . . 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 481 | . 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 396 = wceq 1541 βwex 1781 β wcel 2106 βwral 3061 β c0 4322 class class class wbr 5148 βcfv 6543 (class class class)co 7408 Vtxcvtx 28253 PathsOncpthson 28968 ConnGraphcconngr 29436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7411 df-conngr 29437 |
This theorem is referenced by: 1conngr 29444 |
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