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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 18067 and 2oex 8491) that does not have pairwise disjoint hom-sets, proved by this theorem combined with setc2obas 18076. Notably, the empty set ∅ is simultaneously an object (setc2obas 18076), an identity morphism from ∅ to ∅ (setcid 18068 or thincid 48033), and a non-identity morphism from ∅ to 1o. See cat1lem 18078 and cat1 18079 for a more general statement. This category is also thin (setc2othin 48056), and therefore is "equivalent" to a preorder (actually a partial order). See prsthinc 48054 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 6772 | . . 3 ⊢ ∅:∅⟶∅ | |
2 | setc2ohom.c | . . . . 5 ⊢ 𝐶 = (SetCat‘2o) | |
3 | 2oex 8491 | . . . . . 6 ⊢ 2o ∈ V | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (⊤ → 2o ∈ V) |
5 | setc2ohom.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
6 | 0ex 5301 | . . . . . . . 8 ⊢ ∅ ∈ V | |
7 | 6 | prid1 4762 | . . . . . . 7 ⊢ ∅ ∈ {∅, 1o} |
8 | df2o3 8488 | . . . . . . 7 ⊢ 2o = {∅, 1o} | |
9 | 7, 8 | eleqtrri 2828 | . . . . . 6 ⊢ ∅ ∈ 2o |
10 | 9 | a1i 11 | . . . . 5 ⊢ (⊤ → ∅ ∈ 2o) |
11 | 2, 4, 5, 10, 10 | elsetchom 18063 | . . . 4 ⊢ (⊤ → (∅ ∈ (∅𝐻∅) ↔ ∅:∅⟶∅)) |
12 | 11 | mptru 1541 | . . 3 ⊢ (∅ ∈ (∅𝐻∅) ↔ ∅:∅⟶∅) |
13 | 1, 12 | mpbir 230 | . 2 ⊢ ∅ ∈ (∅𝐻∅) |
14 | f0 6772 | . . 3 ⊢ ∅:∅⟶1o | |
15 | 1oex 8490 | . . . . . . . 8 ⊢ 1o ∈ V | |
16 | 15 | prid2 4763 | . . . . . . 7 ⊢ 1o ∈ {∅, 1o} |
17 | 16, 8 | eleqtrri 2828 | . . . . . 6 ⊢ 1o ∈ 2o |
18 | 17 | a1i 11 | . . . . 5 ⊢ (⊤ → 1o ∈ 2o) |
19 | 2, 4, 5, 10, 18 | elsetchom 18063 | . . . 4 ⊢ (⊤ → (∅ ∈ (∅𝐻1o) ↔ ∅:∅⟶1o)) |
20 | 19 | mptru 1541 | . . 3 ⊢ (∅ ∈ (∅𝐻1o) ↔ ∅:∅⟶1o) |
21 | 14, 20 | mpbir 230 | . 2 ⊢ ∅ ∈ (∅𝐻1o) |
22 | 13, 21 | elini 4189 | 1 ⊢ ∅ ∈ ((∅𝐻∅) ∩ (∅𝐻1o)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 Vcvv 3470 ∩ cin 3944 ∅c0 4318 {cpr 4626 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 1oc1o 8473 2oc2o 8474 Hom chom 17237 SetCatcsetc 18057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-struct 17109 df-slot 17144 df-ndx 17156 df-base 17174 df-hom 17250 df-cco 17251 df-setc 18058 |
This theorem is referenced by: (None) |
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