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Theorem 0catg 15536
 Description: Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
Assertion
Ref Expression
0catg

Proof of Theorem 0catg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 462 . 2
2 eqidd 2429 . 2
3 eqidd 2429 . 2 comp comp
4 simpl 458 . 2
5 noel 3708 . . . 4
65pm2.21i 134 . . 3
8 simpr1 1011 . . 3
95pm2.21i 134 . . 3 comp
108, 9syl 17 . 2 comp
11 simpr1 1011 . . 3
125pm2.21i 134 . . 3 comp
1311, 12syl 17 . 2 comp
14 simp21 1038 . . 3
155pm2.21i 134 . . 3 comp
1614, 15syl 17 . 2 comp
17 simp2ll 1072 . . 3
185pm2.21i 134 . . 3 comp comp comp comp
1917, 18syl 17 . 2 comp comp comp comp
201, 2, 3, 4, 7, 10, 13, 16, 19iscatd 15522 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   w3a 982   wceq 1437   wcel 1872  c0 3704  cop 3947  cfv 5544  (class class class)co 6249  cbs 15064   chom 15144  compcco 15145  ccat 15513 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-nul 4498 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-iota 5508  df-fv 5552  df-ov 6252  df-cat 15517 This theorem is referenced by:  0cat  15537
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