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Theorem uvtx01vtx 25299
 Description: If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. This theorem could have been stated UnivVertex , but a lot of auxiliary theorems would have been needed. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
uvtx01vtx UnivVertex
Distinct variable group:   ,

Proof of Theorem uvtx01vtx
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 4528 . . . . 5
2 isuvtx 25295 . . . . 5 UnivVertex
31, 2mpan2 685 . . . 4 UnivVertex
43neeq1d 2702 . . 3 UnivVertex
5 rabn0 3755 . . . 4
65a1i 11 . . 3
7 df-rex 2762 . . . 4
8 noel 3726 . . . . . . . . . 10
9 rn0 5092 . . . . . . . . . . 11
109eleq2i 2541 . . . . . . . . . 10
118, 10mtbir 306 . . . . . . . . 9
1211ralf0 3867 . . . . . . . 8
1312a1i 11 . . . . . . 7
1413anbi2d 718 . . . . . 6
15 ssdif0 3741 . . . . . . . . 9
1615a1i 11 . . . . . . . 8
1716bicomd 206 . . . . . . 7
1817anbi2d 718 . . . . . 6
19 sssn 4122 . . . . . . . . 9
2019anbi2i 708 . . . . . . . 8
21 n0i 3727 . . . . . . . . . . . 12
2221pm2.21d 109 . . . . . . . . . . 11
2322imp 436 . . . . . . . . . 10
24 simpr 468 . . . . . . . . . 10
2523, 24jaodan 802 . . . . . . . . 9
26 ssnid 3989 . . . . . . . . . . 11
27 eleq2 2538 . . . . . . . . . . 11
2826, 27mpbiri 241 . . . . . . . . . 10
29 olc 391 . . . . . . . . . 10
3028, 29jca 541 . . . . . . . . 9
3125, 30impbii 192 . . . . . . . 8
3220, 31bitri 257 . . . . . . 7
3332a1i 11 . . . . . 6
3414, 18, 333bitrd 287 . . . . 5
3534exbidv 1776 . . . 4
367, 35syl5bb 265 . . 3
374, 6, 363bitrd 287 . 2 UnivVertex
38 id 22 . . . . . . 7
3938intnanrd 931 . . . . . 6
40 df-uvtx 25229 . . . . . . 7 UnivVertex
4140mpt2ndm0 6529 . . . . . 6 UnivVertex
4239, 41syl 17 . . . . 5 UnivVertex
4342notnotd 128 . . . 4 UnivVertex
44 df-ne 2643 . . . 4 UnivVertex UnivVertex
4543, 44sylnibr 312 . . 3 UnivVertex
46 snex 4641 . . . . . 6
47 eleq1 2537 . . . . . 6
4846, 47mpbiri 241 . . . . 5
4948exlimiv 1784 . . . 4
5049con3i 142 . . 3
5145, 502falsed 358 . 2 UnivVertex
5237, 51pm2.61i 169 1 UnivVertex
 Colors of variables: wff setvar class Syntax hints:   wn 3   wb 189   wo 375   wa 376   wceq 1452  wex 1671   wcel 1904   wne 2641  wral 2756  wrex 2757  crab 2760  cvv 3031   cdif 3387   wss 3390  c0 3722  csn 3959  cpr 3961   crn 4840  (class class class)co 6308   UnivVertex cuvtx 25226 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-uvtx 25229 This theorem is referenced by: (None)
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