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Theorem 0cat 16959
 Description: The empty set is a category, the empty category, see example 3.3(4.c) in [Adamek] p. 24. (Contributed by Mario Carneiro, 3-Jan-2017.)
Assertion
Ref Expression
0cat ∅ ∈ Cat

Proof of Theorem 0cat
StepHypRef Expression
1 0ex 5197 . 2 ∅ ∈ V
2 base0 16536 . 2 ∅ = (Base‘∅)
3 0catg 16958 . 2 ((∅ ∈ V ∧ ∅ = (Base‘∅)) → ∅ ∈ Cat)
41, 2, 3mp2an 691 1 ∅ ∈ Cat
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2115  Vcvv 3480  ∅c0 4276  ‘cfv 6343  Basecbs 16483  Catccat 16935 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-slot 16487  df-base 16489  df-cat 16939 This theorem is referenced by: (None)
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