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Mirrors > Home > MPE Home > Th. List > Mathboxes > clsk1independent | Structured version Visualization version Unicode version |
Description: For generalized closure functions, property K1 (isotony) is independent of the properties K0, K2, K3, K4. This contradicts a claim which appears in preprints of Table 2 in Bärbel M. R. Stadler and Peter F. Stadler. "Generalized Topological Spaces in Evolutionary Theory and Combinatorial Chemistry." J. Chem. Inf. Comput. Sci., 42:577-585, 2002. Proceedings MCC 2001, Dubrovnik. The same table row implying K1 follows from the other four appears in the supplemental materials Bärbel M. R. Stadler and Peter F. Stadler. "Basic Properties of Closure Spaces" 2001 on page 12. (Contributed by RP, 5-Jul-2021.) |
Ref | Expression |
---|---|
clsnim.k0 | |
clsnim.k1 | |
clsnim.k2 | |
clsnim.k3 | |
clsnim.k4 |
Ref | Expression |
---|---|
clsk1independent |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3on 7570 | . . 3 | |
2 | 1 | elexi 3213 | . 2 |
3 | eqid 2622 | . . . . 5 | |
4 | notnotr 125 | . . . . . . . . . . 11 | |
5 | 4 | a1i 11 | . . . . . . . . . 10 |
6 | sssucid 5802 | . . . . . . . . . . . . 13 | |
7 | 2on 7568 | . . . . . . . . . . . . . . 15 | |
8 | 7 | elexi 3213 | . . . . . . . . . . . . . 14 |
9 | 8 | elpw 4164 | . . . . . . . . . . . . 13 |
10 | 6, 9 | mpbir 221 | . . . . . . . . . . . 12 |
11 | df2o3 7573 | . . . . . . . . . . . 12 | |
12 | df-3o 7562 | . . . . . . . . . . . . . 14 | |
13 | 12 | eqcomi 2631 | . . . . . . . . . . . . 13 |
14 | 13 | pweqi 4162 | . . . . . . . . . . . 12 |
15 | 10, 11, 14 | 3eltr3i 2713 | . . . . . . . . . . 11 |
16 | 15 | 2a1i 12 | . . . . . . . . . 10 |
17 | 5, 16 | jcad 555 | . . . . . . . . 9 |
18 | 17 | con1d 139 | . . . . . . . 8 |
19 | 18 | anc2ri 581 | . . . . . . 7 |
20 | 19 | orrd 393 | . . . . . 6 |
21 | ifel 4129 | . . . . . 6 | |
22 | 20, 21 | sylibr 224 | . . . . 5 |
23 | 3, 22 | fmpti 6383 | . . . 4 |
24 | 2 | pwex 4848 | . . . . 5 |
25 | 24, 24 | elmap 7886 | . . . 4 |
26 | 23, 25 | mpbir 221 | . . 3 |
27 | 3 | clsk1indlem0 38339 | . . . . . 6 |
28 | 3 | clsk1indlem2 38340 | . . . . . 6 |
29 | 27, 28 | pm3.2i 471 | . . . . 5 |
30 | 3 | clsk1indlem3 38341 | . . . . . 6 |
31 | 3 | clsk1indlem4 38342 | . . . . . 6 |
32 | 30, 31 | pm3.2i 471 | . . . . 5 |
33 | 29, 32 | pm3.2i 471 | . . . 4 |
34 | 3 | clsk1indlem1 38343 | . . . 4 |
35 | 33, 34 | pm3.2i 471 | . . 3 |
36 | fveq1 6190 | . . . . . . . 8 | |
37 | 36 | eqeq1d 2624 | . . . . . . 7 |
38 | fveq1 6190 | . . . . . . . . 9 | |
39 | 38 | sseq2d 3633 | . . . . . . . 8 |
40 | 39 | ralbidv 2986 | . . . . . . 7 |
41 | 37, 40 | anbi12d 747 | . . . . . 6 |
42 | fveq1 6190 | . . . . . . . . 9 | |
43 | fveq1 6190 | . . . . . . . . . 10 | |
44 | 38, 43 | uneq12d 3768 | . . . . . . . . 9 |
45 | 42, 44 | sseq12d 3634 | . . . . . . . 8 |
46 | 45 | 2ralbidv 2989 | . . . . . . 7 |
47 | id 22 | . . . . . . . . . 10 | |
48 | 47, 38 | fveq12d 6197 | . . . . . . . . 9 |
49 | 48, 38 | eqeq12d 2637 | . . . . . . . 8 |
50 | 49 | ralbidv 2986 | . . . . . . 7 |
51 | 46, 50 | anbi12d 747 | . . . . . 6 |
52 | 41, 51 | anbi12d 747 | . . . . 5 |
53 | rexnal2 3043 | . . . . . 6 | |
54 | pm4.61 442 | . . . . . . . 8 | |
55 | 38, 43 | sseq12d 3634 | . . . . . . . . . 10 |
56 | 55 | notbid 308 | . . . . . . . . 9 |
57 | 56 | anbi2d 740 | . . . . . . . 8 |
58 | 54, 57 | syl5bb 272 | . . . . . . 7 |
59 | 58 | 2rexbidv 3057 | . . . . . 6 |
60 | 53, 59 | syl5bbr 274 | . . . . 5 |
61 | 52, 60 | anbi12d 747 | . . . 4 |
62 | 61 | rspcev 3309 | . . 3 |
63 | 26, 35, 62 | mp2an 708 | . 2 |
64 | pweq 4161 | . . . . . 6 | |
65 | 64, 64 | oveq12d 6668 | . . . . 5 |
66 | pm4.61 442 | . . . . . 6 | |
67 | clsnim.k0 | . . . . . . . . . 10 | |
68 | 67 | a1i 11 | . . . . . . . . 9 |
69 | clsnim.k2 | . . . . . . . . . 10 | |
70 | 64 | raleqdv 3144 | . . . . . . . . . 10 |
71 | 69, 70 | syl5bb 272 | . . . . . . . . 9 |
72 | 68, 71 | anbi12d 747 | . . . . . . . 8 |
73 | clsnim.k3 | . . . . . . . . . 10 | |
74 | 64 | raleqdv 3144 | . . . . . . . . . . 11 |
75 | 64, 74 | raleqbidv 3152 | . . . . . . . . . 10 |
76 | 73, 75 | syl5bb 272 | . . . . . . . . 9 |
77 | clsnim.k4 | . . . . . . . . . 10 | |
78 | 64 | raleqdv 3144 | . . . . . . . . . 10 |
79 | 77, 78 | syl5bb 272 | . . . . . . . . 9 |
80 | 76, 79 | anbi12d 747 | . . . . . . . 8 |
81 | 72, 80 | anbi12d 747 | . . . . . . 7 |
82 | clsnim.k1 | . . . . . . . . 9 | |
83 | 64 | raleqdv 3144 | . . . . . . . . . 10 |
84 | 64, 83 | raleqbidv 3152 | . . . . . . . . 9 |
85 | 82, 84 | syl5bb 272 | . . . . . . . 8 |
86 | 85 | notbid 308 | . . . . . . 7 |
87 | 81, 86 | anbi12d 747 | . . . . . 6 |
88 | 66, 87 | syl5bb 272 | . . . . 5 |
89 | 65, 88 | rexeqbidv 3153 | . . . 4 |
90 | 89 | rspcev 3309 | . . 3 |
91 | rexnal2 3043 | . . . 4 | |
92 | ralv 3219 | . . . 4 | |
93 | 91, 92 | xchbinx 324 | . . 3 |
94 | 90, 93 | sylib 208 | . 2 |
95 | 2, 63, 94 | mp2an 708 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 wal 1481 wceq 1483 wcel 1990 wral 2912 wrex 2913 cvv 3200 cun 3572 wss 3574 c0 3915 cif 4086 cpw 4158 csn 4177 cpr 4179 cmpt 4729 con0 5723 csuc 5725 wf 5884 cfv 5888 (class class class)co 6650 c1o 7553 c2o 7554 c3o 7555 cmap 7857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-reg 8497 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-xor 1465 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1o 7560 df-2o 7561 df-3o 7562 df-map 7859 |
This theorem is referenced by: (None) |
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