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Theorem cusgra1v 25181
 Description: A graph with one vertex (and therefore no edges) is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
Assertion
Ref Expression
cusgra1v ComplUSGrph

Proof of Theorem cusgra1v
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 4660 . . . 4
2 usgra0 25089 . . . 4 USGrph
31, 2mp1i 13 . . 3 USGrph
4 ral0 3903 . . . 4
5 sneq 4007 . . . . . . . 8
65difeq2d 3584 . . . . . . 7
7 difid 3864 . . . . . . 7
86, 7syl6eq 2480 . . . . . 6
9 preq2 4078 . . . . . . 7
109eleq1d 2492 . . . . . 6
118, 10raleqbidv 3040 . . . . 5
1211ralsng 4032 . . . 4
134, 12mpbiri 237 . . 3
14 0ex 4554 . . . 4
15 iscusgra 25176 . . . 4 ComplUSGrph USGrph
161, 14, 15mp2an 677 . . 3 ComplUSGrph USGrph
173, 13, 16sylanbrc 669 . 2 ComplUSGrph
18 snprc 4061 . . 3
19 cusgra0v 25180 . . . 4 ComplUSGrph
20 breq1 4424 . . . 4 ComplUSGrph ComplUSGrph
2119, 20mpbiri 237 . . 3 ComplUSGrph
2218, 21sylbi 199 . 2 ComplUSGrph
2317, 22pm2.61i 168 1 ComplUSGrph
 Colors of variables: wff setvar class Syntax hints:   wn 3   wb 188   wa 371   wceq 1438   wcel 1869  wral 2776  cvv 3082   cdif 3434  c0 3762  csn 3997  cpr 3999   class class class wbr 4421   crn 4852   USGrph cusg 25049   ComplUSGrph ccusgra 25138 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-br 4422  df-opab 4481  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-usgra 25052  df-cusgra 25141 This theorem is referenced by: (None)
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