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Mirrors > Home > MPE Home > Th. List > elfzuzb | Structured version Visualization version Unicode version |
Description: Membership in a finite set of sequential integers in terms of sets of upper integers. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
elfzuzb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3an 1039 | . . 3 | |
2 | an6 1408 | . . 3 | |
3 | df-3an 1039 | . . . . 5 | |
4 | anandir 872 | . . . . 5 | |
5 | ancom 466 | . . . . . 6 | |
6 | 5 | anbi2i 730 | . . . . 5 |
7 | 3, 4, 6 | 3bitri 286 | . . . 4 |
8 | 7 | anbi1i 731 | . . 3 |
9 | 1, 2, 8 | 3bitr4ri 293 | . 2 |
10 | elfz2 12333 | . 2 | |
11 | eluz2 11693 | . . 3 | |
12 | eluz2 11693 | . . 3 | |
13 | 11, 12 | anbi12i 733 | . 2 |
14 | 9, 10, 13 | 3bitr4i 292 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 w3a 1037 wcel 1990 class class class wbr 4653 cfv 5888 (class class class)co 6650 cle 10075 cz 11377 cuz 11687 cfz 12326 |
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-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 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-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 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-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-neg 10269 df-z 11378 df-uz 11688 df-fz 12327 |
This theorem is referenced by: eluzfz 12337 elfzuz 12338 elfzuz3 12339 elfzuz2 12346 peano2fzr 12354 fzsplit2 12366 fzass4 12379 fzss1 12380 fzss2 12381 ssfzunsnext 12386 fzp1elp1 12394 fznn 12408 elfz2nn0 12431 elfzofz 12485 fzosplitsnm1 12542 fzofzp1b 12566 fzosplitsn 12576 seqcl2 12819 seqfveq2 12823 monoord 12831 seqid2 12847 bcn1 13100 fz1isolem 13245 seqcoll 13248 ccatrn 13372 swrds1 13451 swrdccat1 13457 swrdccat2 13458 spllen 13505 splfv2a 13507 splval2 13508 caubnd 14098 isercolllem2 14396 isercolllem3 14397 summolem2a 14446 fsum0diag2 14515 climcndslem1 14581 mertenslem1 14616 prodmolem2a 14664 vdwlem2 15686 vdwlem8 15692 gexcl3 18002 efginvrel2 18140 efgredleme 18156 efgcpbllemb 18168 1stckgenlem 21356 imasdsf1olem 22178 iscmet3lem1 23089 dvtaylp 24124 mtest 24158 ppisval 24830 ppisval2 24831 chtdif 24884 ppidif 24889 logfaclbnd 24947 bposlem4 25012 dchrisumlem2 25179 pntpbnd1 25275 fzsplit3 29553 mettrifi 33553 smonoord 41341 |
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