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Theorem frgra0v 25713
 Description: Any graph with no vertex is a friendship graph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
Assertion
Ref Expression
frgra0v FriendGrph

Proof of Theorem frgra0v
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frisusgra 25712 . . 3 FriendGrph USGrph
2 usgra0v 25090 . . 3 USGrph
31, 2sylib 200 . 2 FriendGrph
42biimpri 210 . . 3 USGrph
5 ral0 3903 . . . 4
65a1i 11 . . 3
7 0ex 4554 . . . 4
8 eleq1 2495 . . . . 5
97, 8mpbiri 237 . . . 4
10 isfrgra 25710 . . . 4 FriendGrph USGrph
117, 9, 10sylancr 668 . . 3 FriendGrph USGrph
124, 6, 11mpbir2and 931 . 2 FriendGrph
133, 12impbii 191 1 FriendGrph
 Colors of variables: wff setvar class Syntax hints:   wb 188   wa 371   wceq 1438   wcel 1869  wral 2776  wreu 2778  cvv 3082   cdif 3434   wss 3437  c0 3762  csn 3997  cpr 3999   class class class wbr 4421   crn 4852   USGrph cusg 25049   FriendGrph cfrgra 25708 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-reu 2783  df-rab 2785  df-v 3084  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-frgra 25709 This theorem is referenced by:  frgra0  25714
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