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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 17545 and 2oex 8197) that does not have pairwise disjoint hom-sets, proved by this theorem combined with setc2obas 17554. Notably, the empty set ∅ is simultaneously an object (setc2obas 17554) , an identity morphism from ∅ to ∅ (setcid 17546 or thincid 45930) , and a non-identity morphism from ∅ to 1o. See cat1lem 17556 and cat1 17557 for a more general statement. This category is also thin (setc2othin 45953), and therefore is "equivalent" to a preorder (actually a partial order). See prsthinc 45951 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 6578 | . . 3 ⊢ ∅:∅⟶∅ | |
2 | setc2ohom.c | . . . . 5 ⊢ 𝐶 = (SetCat‘2o) | |
3 | 2oex 8197 | . . . . . 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 4664 | . . . . . . 7 ⊢ ∅ ∈ {∅, 1o} |
8 | df2o3 8195 | . . . . . . 7 ⊢ 2o = {∅, 1o} | |
9 | 7, 8 | eleqtrri 2830 | . . . . . 6 ⊢ ∅ ∈ 2o |
10 | 9 | a1i 11 | . . . . 5 ⊢ (⊤ → ∅ ∈ 2o) |
11 | 2, 4, 5, 10, 10 | elsetchom 17541 | . . . 4 ⊢ (⊤ → (∅ ∈ (∅𝐻∅) ↔ ∅:∅⟶∅)) |
12 | 11 | mptru 1550 | . . 3 ⊢ (∅ ∈ (∅𝐻∅) ↔ ∅:∅⟶∅) |
13 | 1, 12 | mpbir 234 | . 2 ⊢ ∅ ∈ (∅𝐻∅) |
14 | f0 6578 | . . 3 ⊢ ∅:∅⟶1o | |
15 | 1oex 8193 | . . . . . . . 8 ⊢ 1o ∈ V | |
16 | 15 | prid2 4665 | . . . . . . 7 ⊢ 1o ∈ {∅, 1o} |
17 | 16, 8 | eleqtrri 2830 | . . . . . 6 ⊢ 1o ∈ 2o |
18 | 17 | a1i 11 | . . . . 5 ⊢ (⊤ → 1o ∈ 2o) |
19 | 2, 4, 5, 10, 18 | elsetchom 17541 | . . . 4 ⊢ (⊤ → (∅ ∈ (∅𝐻1o) ↔ ∅:∅⟶1o)) |
20 | 19 | mptru 1550 | . . 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 1543 ⊤wtru 1544 ∈ wcel 2112 Vcvv 3398 ∩ cin 3852 ∅c0 4223 {cpr 4529 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 1oc1o 8173 2oc2o 8174 Hom chom 16760 SetCatcsetc 17535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-hom 16773 df-cco 16774 df-setc 17536 |
This theorem is referenced by: (None) |
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