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Mirrors > Home > MPE Home > Th. List > 0conngr | 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 |
---|---|
0conngr | β’ β β ConnGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ral0 4471 | . 2 β’ βπ β β βπ β β βπβπ π(π(PathsOnββ )π)π | |
2 | 0ex 5265 | . . 3 β’ β β V | |
3 | vtxval0 27993 | . . . . 5 β’ (Vtxββ ) = β | |
4 | 3 | eqcomi 2746 | . . . 4 β’ β = (Vtxββ ) |
5 | 4 | isconngr 29136 | . . 3 β’ (β β V β (β β ConnGraph β βπ β β βπ β β βπβπ π(π(PathsOnββ )π)π)) |
6 | 2, 5 | ax-mp 5 | . 2 β’ (β β ConnGraph β βπ β β βπ β β βπβπ π(π(PathsOnββ )π)π) |
7 | 1, 6 | mpbir 230 | 1 β’ β β ConnGraph |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 βwex 1782 β wcel 2107 βwral 3065 Vcvv 3446 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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-1cn 11110 ax-addcl 11112 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-nn 12155 df-slot 17055 df-ndx 17067 df-base 17085 df-vtx 27952 df-conngr 29134 |
This theorem is referenced by: (None) |
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