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Theorem 0catg 17639
Description: Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
Assertion
Ref Expression
0catg ((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ Cat)

Proof of Theorem 0catg
Dummy variables 𝑓 𝑔 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 484 . 2 ((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) → ∅ = (Base‘𝐶))
2 eqidd 2732 . 2 ((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) → (Hom ‘𝐶) = (Hom ‘𝐶))
3 eqidd 2732 . 2 ((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) → (comp‘𝐶) = (comp‘𝐶))
4 simpl 482 . 2 ((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶𝑉)
5 noel 4330 . . . 4 ¬ 𝑥 ∈ ∅
65pm2.21i 119 . . 3 (𝑥 ∈ ∅ → ∅ ∈ (𝑥(Hom ‘𝐶)𝑥))
76adantl 481 . 2 (((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) ∧ 𝑥 ∈ ∅) → ∅ ∈ (𝑥(Hom ‘𝐶)𝑥))
8 simpr1 1193 . . 3 (((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))) → 𝑥 ∈ ∅)
95pm2.21i 119 . . 3 (𝑥 ∈ ∅ → (∅(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
108, 9syl 17 . 2 (((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))) → (∅(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
11 simpr1 1193 . . 3 (((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑥 ∈ ∅)
125pm2.21i 119 . . 3 (𝑥 ∈ ∅ → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)∅) = 𝑓)
1311, 12syl 17 . 2 (((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)∅) = 𝑓)
14 simp21 1205 . . 3 (((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥 ∈ ∅)
155pm2.21i 119 . . 3 (𝑥 ∈ ∅ → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
1614, 15syl 17 . 2 (((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
17 simp2ll 1239 . . 3 (((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅) ∧ (𝑧 ∈ ∅ ∧ 𝑤 ∈ ∅)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤))) → 𝑥 ∈ ∅)
185pm2.21i 119 . . 3 (𝑥 ∈ ∅ → (((⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = ((⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))
1917, 18syl 17 . 2 (((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅) ∧ (𝑧 ∈ ∅ ∧ 𝑤 ∈ ∅)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤))) → (((⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = ((⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))
201, 2, 3, 4, 7, 10, 13, 16, 19iscatd 17624 1 ((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ Cat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1540  wcel 2105  c0 4322  cop 4634  cfv 6543  (class class class)co 7412  Basecbs 17151  Hom chom 17215  compcco 17216  Catccat 17615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415  df-cat 17619
This theorem is referenced by:  0cat  17640  0thincg  47834
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