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Mirrors > Home > MPE Home > Th. List > uvtxa01vtx | Structured version Visualization version Unicode version |
Description: If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.) |
Ref | Expression |
---|---|
uvtxael.v | Vtx |
isuvtxa.e | Edg |
Ref | Expression |
---|---|
uvtxa01vtx | UnivVtx |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uvtxael.v | . . 3 Vtx | |
2 | isuvtxa.e | . . 3 Edg | |
3 | 1, 2 | uvtxa01vtx0 26297 | . 2 UnivVtx |
4 | fvprc 6185 | . . . . 5 UnivVtx | |
5 | 4 | adantr 481 | . . . 4 UnivVtx |
6 | nne 2798 | . . . 4 UnivVtx UnivVtx | |
7 | 5, 6 | sylibr 224 | . . 3 UnivVtx |
8 | fvprc 6185 | . . . . 5 Vtx | |
9 | 1 | eqcomi 2631 | . . . . . . 7 Vtx |
10 | 9 | eqeq1i 2627 | . . . . . 6 Vtx |
11 | 0ne1 11088 | . . . . . . . 8 | |
12 | 11 | neii 2796 | . . . . . . 7 |
13 | fveq2 6191 | . . . . . . . . 9 | |
14 | hash0 13158 | . . . . . . . . 9 | |
15 | 13, 14 | syl6eq 2672 | . . . . . . . 8 |
16 | 15 | eqeq1d 2624 | . . . . . . 7 |
17 | 12, 16 | mtbiri 317 | . . . . . 6 |
18 | 10, 17 | sylbi 207 | . . . . 5 Vtx |
19 | 8, 18 | syl 17 | . . . 4 |
20 | 19 | adantr 481 | . . 3 |
21 | 7, 20 | 2falsed 366 | . 2 UnivVtx |
22 | 3, 21 | pm2.61ian 831 | 1 UnivVtx |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wne 2794 cvv 3200 c0 3915 cfv 5888 cc0 9936 c1 9937 chash 13117 Vtxcvtx 25874 Edgcedg 25939 UnivVtxcuvtxa 26225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118 df-nbgr 26228 df-uvtxa 26230 |
This theorem is referenced by: (None) |
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