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Mathbox for Jim Kingdon |
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Mirrors > Home > ILE Home > Th. List > Mathboxes > 012of | Unicode version |
Description: Mapping zero and one
between ![]() ![]() |
Ref | Expression |
---|---|
012of.g |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
012of |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 012of.g |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | 1 | frechashgf1o 10305 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() |
3 | f1ocnv 5420 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | f1of 5407 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 2, 3, 4 | mp2b 8 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | 0nn0 9084 |
. . . . 5
![]() ![]() ![]() ![]() | |
7 | 1nn0 9085 |
. . . . 5
![]() ![]() ![]() ![]() | |
8 | prssi 3710 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 6, 7, 8 | mp2an 423 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | fssres 5338 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 5, 9, 10 | mp2an 423 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | ffn 5312 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 11, 12 | ax-mp 5 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | fvres 5485 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | elpri 3579 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | fveq2 5461 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 0zd 9158 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | 17, 1 | frec2uz0d 10276 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 18 | mptru 1341 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | peano1 4547 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() | |
21 | f1ocnvfv 5720 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | 2, 20, 21 | mp2an 423 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 19, 22 | ax-mp 5 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 0lt2o 6378 |
. . . . . . . 8
![]() ![]() ![]() ![]() | |
25 | 23, 24 | eqeltri 2227 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 16, 25 | eqeltrdi 2245 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | fveq2 5461 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | df-1o 6353 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() | |
29 | 28 | fveq2i 5464 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 20 | a1i 9 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 17, 1, 30 | frec2uzsucd 10278 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 31 | mptru 1341 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 19 | oveq1i 5824 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 0p1e1 8926 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
35 | 33, 34 | eqtri 2175 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | 29, 32, 35 | 3eqtri 2179 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
37 | 1onn 6456 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() | |
38 | f1ocnvfv 5720 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
39 | 2, 37, 38 | mp2an 423 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 36, 39 | ax-mp 5 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | 1lt2o 6379 |
. . . . . . . 8
![]() ![]() ![]() ![]() | |
42 | 40, 41 | eqeltri 2227 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
43 | 27, 42 | eqeltrdi 2245 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
44 | 26, 43 | jaoi 706 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
45 | 15, 44 | syl 14 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46 | 14, 45 | eqeltrd 2231 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
47 | 46 | rgen 2507 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | ffnfv 5618 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
49 | 13, 47, 48 | mpbir2an 927 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-coll 4075 ax-sep 4078 ax-nul 4086 ax-pow 4130 ax-pr 4164 ax-un 4388 ax-setind 4490 ax-iinf 4541 ax-cnex 7802 ax-resscn 7803 ax-1cn 7804 ax-1re 7805 ax-icn 7806 ax-addcl 7807 ax-addrcl 7808 ax-mulcl 7809 ax-addcom 7811 ax-addass 7813 ax-distr 7815 ax-i2m1 7816 ax-0lt1 7817 ax-0id 7819 ax-rnegex 7820 ax-cnre 7822 ax-pre-ltirr 7823 ax-pre-ltwlin 7824 ax-pre-lttrn 7825 ax-pre-ltadd 7827 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-nel 2420 df-ral 2437 df-rex 2438 df-reu 2439 df-rab 2441 df-v 2711 df-sbc 2934 df-csb 3028 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-nul 3391 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-int 3804 df-iun 3847 df-br 3962 df-opab 4022 df-mpt 4023 df-tr 4059 df-id 4248 df-iord 4321 df-on 4323 df-ilim 4324 df-suc 4326 df-iom 4544 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-rn 4590 df-res 4591 df-ima 4592 df-iota 5128 df-fun 5165 df-fn 5166 df-f 5167 df-f1 5168 df-fo 5169 df-f1o 5170 df-fv 5171 df-riota 5770 df-ov 5817 df-oprab 5818 df-mpo 5819 df-recs 6242 df-frec 6328 df-1o 6353 df-2o 6354 df-pnf 7893 df-mnf 7894 df-xr 7895 df-ltxr 7896 df-le 7897 df-sub 8027 df-neg 8028 df-inn 8813 df-n0 9070 df-z 9147 df-uz 9419 |
This theorem is referenced by: isomninnlem 13542 iswomninnlem 13561 ismkvnnlem 13564 |
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