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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 18047 and 2oex 8478) that does not have pairwise disjoint hom-sets, proved by this theorem combined with setc2obas 18056. Notably, the empty set ∅ is simultaneously an object (setc2obas 18056), an identity morphism from ∅ to ∅ (setcid 18048 or thincid 47927), and a non-identity morphism from ∅ to 1o. See cat1lem 18058 and cat1 18059 for a more general statement. This category is also thin (setc2othin 47950), and therefore is "equivalent" to a preorder (actually a partial order). See prsthinc 47948 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 6766 | . . 3 ⊢ ∅:∅⟶∅ | |
2 | setc2ohom.c | . . . . 5 ⊢ 𝐶 = (SetCat‘2o) | |
3 | 2oex 8478 | . . . . . 6 ⊢ 2o ∈ V | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (⊤ → 2o ∈ V) |
5 | setc2ohom.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
6 | 0ex 5300 | . . . . . . . 8 ⊢ ∅ ∈ V | |
7 | 6 | prid1 4761 | . . . . . . 7 ⊢ ∅ ∈ {∅, 1o} |
8 | df2o3 8475 | . . . . . . 7 ⊢ 2o = {∅, 1o} | |
9 | 7, 8 | eleqtrri 2826 | . . . . . 6 ⊢ ∅ ∈ 2o |
10 | 9 | a1i 11 | . . . . 5 ⊢ (⊤ → ∅ ∈ 2o) |
11 | 2, 4, 5, 10, 10 | elsetchom 18043 | . . . 4 ⊢ (⊤ → (∅ ∈ (∅𝐻∅) ↔ ∅:∅⟶∅)) |
12 | 11 | mptru 1540 | . . 3 ⊢ (∅ ∈ (∅𝐻∅) ↔ ∅:∅⟶∅) |
13 | 1, 12 | mpbir 230 | . 2 ⊢ ∅ ∈ (∅𝐻∅) |
14 | f0 6766 | . . 3 ⊢ ∅:∅⟶1o | |
15 | 1oex 8477 | . . . . . . . 8 ⊢ 1o ∈ V | |
16 | 15 | prid2 4762 | . . . . . . 7 ⊢ 1o ∈ {∅, 1o} |
17 | 16, 8 | eleqtrri 2826 | . . . . . 6 ⊢ 1o ∈ 2o |
18 | 17 | a1i 11 | . . . . 5 ⊢ (⊤ → 1o ∈ 2o) |
19 | 2, 4, 5, 10, 18 | elsetchom 18043 | . . . 4 ⊢ (⊤ → (∅ ∈ (∅𝐻1o) ↔ ∅:∅⟶1o)) |
20 | 19 | mptru 1540 | . . 3 ⊢ (∅ ∈ (∅𝐻1o) ↔ ∅:∅⟶1o) |
21 | 14, 20 | mpbir 230 | . 2 ⊢ ∅ ∈ (∅𝐻1o) |
22 | 13, 21 | elini 4188 | 1 ⊢ ∅ ∈ ((∅𝐻∅) ∩ (∅𝐻1o)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 Vcvv 3468 ∩ cin 3942 ∅c0 4317 {cpr 4625 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 1oc1o 8460 2oc2o 8461 Hom chom 17217 SetCatcsetc 18037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-slot 17124 df-ndx 17136 df-base 17154 df-hom 17230 df-cco 17231 df-setc 18038 |
This theorem is referenced by: (None) |
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