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Theorem 0catg 13605
 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 447 . 2
2 eqidd 2297 . 2
3 eqidd 2297 . 2 comp comp
4 simpl 443 . 2
5 noel 3472 . . . 4
65pm2.21i 123 . . 3
8 simpr1 961 . . 3
95pm2.21i 123 . . 3 comp
108, 9syl 15 . 2 comp
11 simpr1 961 . . 3
125pm2.21i 123 . . 3 comp
1311, 12syl 15 . 2 comp
14 simp21 988 . . 3
155pm2.21i 123 . . 3 comp
1614, 15syl 15 . 2 comp
17 simp2ll 1022 . . 3
185pm2.21i 123 . . 3 comp comp comp comp
1917, 18syl 15 . 2 comp comp comp comp
201, 2, 3, 4, 7, 10, 13, 16, 19iscatd 13591 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3a 934   wceq 1632   wcel 1696  c0 3468  cop 3656  cfv 5271  (class class class)co 5874  cbs 13164   chom 13235  compcco 13236  ccat 13582 This theorem is referenced by:  0cat  13606 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-cat 13586
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