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Mirrors > Home > MPE Home > Th. List > setc2ohom | Structured version Visualization version GIF version |
Description: (SetCat‘2o) is a category (provable from setccat 17469 and 2oex 8160) that does not have pairwise disjoint hom-sets, proved by this theorem combined with setc2obas 17478. Notably, the empty set ∅ is simultaneously an object (setc2obas 17478) , an identity morphism from ∅ to ∅, and a non-identity morphism from ∅ to 1o. See cat1lem 17480 and cat1 17481 for a more general statement. This category is also thin (setc2othin 45848), and therefore is "equivalent" to a preorder (actually a partial order). See prsthinc 45846 for more details on the "equivalence". (Contributed by Zhi Wang, 24-Sep-2024.) |
Ref | Expression |
---|---|
setc2ohom.c | ⊢ 𝐶 = (SetCat‘2o) |
setc2ohom.h | ⊢ 𝐻 = (Hom ‘𝐶) |
Ref | Expression |
---|---|
setc2ohom | ⊢ ∅ ∈ ((∅𝐻∅) ∩ (∅𝐻1o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f0 6569 | . . 3 ⊢ ∅:∅⟶∅ | |
2 | setc2ohom.c | . . . . 5 ⊢ 𝐶 = (SetCat‘2o) | |
3 | 2oex 8160 | . . . . . 6 ⊢ 2o ∈ V | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (⊤ → 2o ∈ V) |
5 | setc2ohom.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
6 | 0ex 5185 | . . . . . . . 8 ⊢ ∅ ∈ V | |
7 | 6 | prid1 4663 | . . . . . . 7 ⊢ ∅ ∈ {∅, 1o} |
8 | df2o3 8158 | . . . . . . 7 ⊢ 2o = {∅, 1o} | |
9 | 7, 8 | eleqtrri 2833 | . . . . . 6 ⊢ ∅ ∈ 2o |
10 | 9 | a1i 11 | . . . . 5 ⊢ (⊤ → ∅ ∈ 2o) |
11 | 2, 4, 5, 10, 10 | elsetchom 17465 | . . . 4 ⊢ (⊤ → (∅ ∈ (∅𝐻∅) ↔ ∅:∅⟶∅)) |
12 | 11 | mptru 1549 | . . 3 ⊢ (∅ ∈ (∅𝐻∅) ↔ ∅:∅⟶∅) |
13 | 1, 12 | mpbir 234 | . 2 ⊢ ∅ ∈ (∅𝐻∅) |
14 | f0 6569 | . . 3 ⊢ ∅:∅⟶1o | |
15 | 1oex 8156 | . . . . . . . 8 ⊢ 1o ∈ V | |
16 | 15 | prid2 4664 | . . . . . . 7 ⊢ 1o ∈ {∅, 1o} |
17 | 16, 8 | eleqtrri 2833 | . . . . . 6 ⊢ 1o ∈ 2o |
18 | 17 | a1i 11 | . . . . 5 ⊢ (⊤ → 1o ∈ 2o) |
19 | 2, 4, 5, 10, 18 | elsetchom 17465 | . . . 4 ⊢ (⊤ → (∅ ∈ (∅𝐻1o) ↔ ∅:∅⟶1o)) |
20 | 19 | mptru 1549 | . . 3 ⊢ (∅ ∈ (∅𝐻1o) ↔ ∅:∅⟶1o) |
21 | 14, 20 | mpbir 234 | . 2 ⊢ ∅ ∈ (∅𝐻1o) |
22 | 13, 21 | elini 4093 | 1 ⊢ ∅ ∈ ((∅𝐻∅) ∩ (∅𝐻1o)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 Vcvv 3400 ∩ cin 3852 ∅c0 4221 {cpr 4528 ⟶wf 6345 ‘cfv 6349 (class class class)co 7182 1oc1o 8136 2oc2o 8137 Hom chom 16691 SetCatcsetc 17459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-2o 8144 df-er 8332 df-map 8451 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-7 11796 df-8 11797 df-9 11798 df-n0 11989 df-z 12075 df-dec 12192 df-uz 12337 df-fz 12994 df-struct 16600 df-ndx 16601 df-slot 16602 df-base 16604 df-hom 16704 df-cco 16705 df-setc 17460 |
This theorem is referenced by: (None) |
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