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Theorem frgra1v 25702
 Description: Any graph with only one vertex is a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
Assertion
Ref Expression
frgra1v USGrph FriendGrph

Proof of Theorem frgra1v
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgrav 25042 . . 3 USGrph
2 simplr 760 . . . . 5 USGrph USGrph
3 ral0 3899 . . . . . 6
4 sneq 4003 . . . . . . . . . . 11
54difeq2d 3580 . . . . . . . . . 10
6 difid 3860 . . . . . . . . . 10
75, 6syl6eq 2477 . . . . . . . . 9
8 preq2 4074 . . . . . . . . . . . 12
98preq1d 4079 . . . . . . . . . . 11
109sseq1d 3488 . . . . . . . . . 10
1110reubidv 3011 . . . . . . . . 9
127, 11raleqbidv 3037 . . . . . . . 8
1312ralsng 4028 . . . . . . 7
1413adantl 467 . . . . . 6 USGrph
153, 14mpbiri 236 . . . . 5 USGrph
16 isfrgra 25694 . . . . . 6 FriendGrph USGrph
1716ad2antrr 730 . . . . 5 USGrph FriendGrph USGrph
182, 15, 17mpbir2and 930 . . . 4 USGrph FriendGrph
1918ex 435 . . 3 USGrph FriendGrph
201, 19mpancom 673 . 2 USGrph FriendGrph
2120impcom 431 1 USGrph FriendGrph
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   wceq 1437   wcel 1867  wral 2773  wreu 2775  cvv 3078   cdif 3430   wss 3433  c0 3758  csn 3993  cpr 3995   class class class wbr 4417   crn 4847   USGrph cusg 25034   FriendGrph cfrgra 25692 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pr 4653 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4477  df-xp 4852  df-rel 4853  df-cnv 4854  df-dm 4856  df-rn 4857  df-usgra 25037  df-frgra 25693 This theorem is referenced by: (None)
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