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Mirrors > Home > MPE Home > Th. List > 0cat | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
0cat | ⊢ ∅ ∈ Cat |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5308 | . 2 ⊢ ∅ ∈ V | |
2 | base0 17149 | . 2 ⊢ ∅ = (Base‘∅) | |
3 | 0catg 17632 | . 2 ⊢ ((∅ ∈ V ∧ ∅ = (Base‘∅)) → ∅ ∈ Cat) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ ∅ ∈ Cat |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∅c0 4323 ‘cfv 6544 Basecbs 17144 Catccat 17608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-1cn 11168 ax-addcl 11170 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-nn 12213 df-slot 17115 df-ndx 17127 df-base 17145 df-cat 17612 |
This theorem is referenced by: (None) |
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