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Mirrors > Home > MPE Home > Th. List > setc2obas | Structured version Visualization version GIF version |
Description: ∅ and 1o are distinct objects in (SetCat‘2o). This combined with setc2ohom 18024 demonstrates that the category does not have pairwise disjoint hom-sets. See also df-cat 17591 and cat1 18026. (Contributed by Zhi Wang, 24-Sep-2024.) |
Ref | Expression |
---|---|
setc2ohom.c | ⊢ 𝐶 = (SetCat‘2o) |
setc2obas.b | ⊢ 𝐵 = (Base‘𝐶) |
Ref | Expression |
---|---|
setc2obas | ⊢ (∅ ∈ 𝐵 ∧ 1o ∈ 𝐵 ∧ 1o ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5297 | . . . 4 ⊢ ∅ ∈ V | |
2 | 1 | prid1 4756 | . . 3 ⊢ ∅ ∈ {∅, 1o} |
3 | setc2obas.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
4 | setc2ohom.c | . . . . . 6 ⊢ 𝐶 = (SetCat‘2o) | |
5 | 2oex 8456 | . . . . . . 7 ⊢ 2o ∈ V | |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (⊤ → 2o ∈ V) |
7 | 4, 6 | setcbas 18007 | . . . . 5 ⊢ (⊤ → 2o = (Base‘𝐶)) |
8 | 7 | mptru 1548 | . . . 4 ⊢ 2o = (Base‘𝐶) |
9 | df2o3 8453 | . . . 4 ⊢ 2o = {∅, 1o} | |
10 | 3, 8, 9 | 3eqtr2i 2765 | . . 3 ⊢ 𝐵 = {∅, 1o} |
11 | 2, 10 | eleqtrri 2831 | . 2 ⊢ ∅ ∈ 𝐵 |
12 | 1oex 8455 | . . . 4 ⊢ 1o ∈ V | |
13 | 12 | prid2 4757 | . . 3 ⊢ 1o ∈ {∅, 1o} |
14 | 13, 10 | eleqtrri 2831 | . 2 ⊢ 1o ∈ 𝐵 |
15 | 1n0 8467 | . 2 ⊢ 1o ≠ ∅ | |
16 | 11, 14, 15 | 3pm3.2i 1339 | 1 ⊢ (∅ ∈ 𝐵 ∧ 1o ∈ 𝐵 ∧ 1o ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1087 = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 ≠ wne 2939 Vcvv 3470 ∅c0 4315 {cpr 4621 ‘cfv 6529 1oc1o 8438 2oc2o 8439 Basecbs 17123 SetCatcsetc 18004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2702 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-cnex 11145 ax-resscn 11146 ax-1cn 11147 ax-icn 11148 ax-addcl 11149 ax-addrcl 11150 ax-mulcl 11151 ax-mulrcl 11152 ax-mulcom 11153 ax-addass 11154 ax-mulass 11155 ax-distr 11156 ax-i2m1 11157 ax-1ne0 11158 ax-1rid 11159 ax-rnegex 11160 ax-rrecex 11161 ax-cnre 11162 ax-pre-lttri 11163 ax-pre-lttrn 11164 ax-pre-ltadd 11165 ax-pre-mulgt0 11166 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-tp 4624 df-op 4626 df-uni 4899 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6286 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-riota 7346 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7836 df-1st 7954 df-2nd 7955 df-frecs 8245 df-wrecs 8276 df-recs 8350 df-rdg 8389 df-1o 8445 df-2o 8446 df-er 8683 df-en 8920 df-dom 8921 df-sdom 8922 df-fin 8923 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11425 df-neg 11426 df-nn 12192 df-2 12254 df-3 12255 df-4 12256 df-5 12257 df-6 12258 df-7 12259 df-8 12260 df-9 12261 df-n0 12452 df-z 12538 df-dec 12657 df-uz 12802 df-fz 13464 df-struct 17059 df-slot 17094 df-ndx 17106 df-base 17124 df-hom 17200 df-cco 17201 df-setc 18005 |
This theorem is referenced by: (None) |
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