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Theorem uhgra0v 24714
 Description: The null graph, with no vertices, is a hypergraph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
Assertion
Ref Expression
uhgra0v UHGrph

Proof of Theorem uhgra0v
StepHypRef Expression
1 uhgrav 24700 . . 3 UHGrph
2 isuhgra 24702 . . . 4 UHGrph
3 eqid 2402 . . . . . 6
4 pw0 4118 . . . . . . . 8
54difeq1i 3556 . . . . . . 7
6 difid 3839 . . . . . . 7
75, 6eqtri 2431 . . . . . 6
83, 7feq23i 5707 . . . . 5
9 f00 5749 . . . . . 6
109simplbi 458 . . . . 5
118, 10sylbi 195 . . . 4
122, 11syl6bi 228 . . 3 UHGrph
131, 12mpcom 34 . 2 UHGrph
14 0ex 4525 . . . 4
15 uhgra0 24713 . . . 4 UHGrph
1614, 15ax-mp 5 . . 3 UHGrph
17 breq2 4398 . . 3 UHGrph UHGrph
1816, 17mpbiri 233 . 2 UHGrph
1913, 18impbii 188 1 UHGrph
 Colors of variables: wff setvar class Syntax hints:   wb 184   wa 367   wceq 1405   wcel 1842  cvv 3058   cdif 3410  c0 3737  cpw 3954  csn 3971   class class class wbr 4394   cdm 4822  wf 5564   UHGrph cuhg 24694 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-fun 5570  df-fn 5571  df-f 5572  df-uhgra 24696 This theorem is referenced by: (None)
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