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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 17800 and 2oex 8308) that does not have pairwise disjoint hom-sets, proved by this theorem combined with setc2obas 17809. Notably, the empty set ∅ is simultaneously an object (setc2obas 17809) , an identity morphism from ∅ to ∅ (setcid 17801 or thincid 46314) , and a non-identity morphism from ∅ to 1o. See cat1lem 17811 and cat1 17812 for a more general statement. This category is also thin (setc2othin 46337), and therefore is "equivalent" to a preorder (actually a partial order). See prsthinc 46335 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 6655 | . . 3 ⊢ ∅:∅⟶∅ | |
2 | setc2ohom.c | . . . . 5 ⊢ 𝐶 = (SetCat‘2o) | |
3 | 2oex 8308 | . . . . . 6 ⊢ 2o ∈ V | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (⊤ → 2o ∈ V) |
5 | setc2ohom.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
6 | 0ex 5231 | . . . . . . . 8 ⊢ ∅ ∈ V | |
7 | 6 | prid1 4698 | . . . . . . 7 ⊢ ∅ ∈ {∅, 1o} |
8 | df2o3 8305 | . . . . . . 7 ⊢ 2o = {∅, 1o} | |
9 | 7, 8 | eleqtrri 2838 | . . . . . 6 ⊢ ∅ ∈ 2o |
10 | 9 | a1i 11 | . . . . 5 ⊢ (⊤ → ∅ ∈ 2o) |
11 | 2, 4, 5, 10, 10 | elsetchom 17796 | . . . 4 ⊢ (⊤ → (∅ ∈ (∅𝐻∅) ↔ ∅:∅⟶∅)) |
12 | 11 | mptru 1546 | . . 3 ⊢ (∅ ∈ (∅𝐻∅) ↔ ∅:∅⟶∅) |
13 | 1, 12 | mpbir 230 | . 2 ⊢ ∅ ∈ (∅𝐻∅) |
14 | f0 6655 | . . 3 ⊢ ∅:∅⟶1o | |
15 | 1oex 8307 | . . . . . . . 8 ⊢ 1o ∈ V | |
16 | 15 | prid2 4699 | . . . . . . 7 ⊢ 1o ∈ {∅, 1o} |
17 | 16, 8 | eleqtrri 2838 | . . . . . 6 ⊢ 1o ∈ 2o |
18 | 17 | a1i 11 | . . . . 5 ⊢ (⊤ → 1o ∈ 2o) |
19 | 2, 4, 5, 10, 18 | elsetchom 17796 | . . . 4 ⊢ (⊤ → (∅ ∈ (∅𝐻1o) ↔ ∅:∅⟶1o)) |
20 | 19 | mptru 1546 | . . 3 ⊢ (∅ ∈ (∅𝐻1o) ↔ ∅:∅⟶1o) |
21 | 14, 20 | mpbir 230 | . 2 ⊢ ∅ ∈ (∅𝐻1o) |
22 | 13, 21 | elini 4127 | 1 ⊢ ∅ ∈ ((∅𝐻∅) ∩ (∅𝐻1o)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1539 ⊤wtru 1540 ∈ wcel 2106 Vcvv 3432 ∩ cin 3886 ∅c0 4256 {cpr 4563 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 1oc1o 8290 2oc2o 8291 Hom chom 16973 SetCatcsetc 17790 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-struct 16848 df-slot 16883 df-ndx 16895 df-base 16913 df-hom 16986 df-cco 16987 df-setc 17791 |
This theorem is referenced by: (None) |
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