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Mirrors > Home > ILE Home > Th. List > elfzuzb | 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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3an 886 |
. . 3
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2 | an6 1215 |
. . 3
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3 | df-3an 886 |
. . . . 5
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4 | anandir 525 |
. . . . 5
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5 | ancom 253 |
. . . . . 6
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6 | 5 | anbi2i 430 |
. . . . 5
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7 | 3, 4, 6 | 3bitri 195 |
. . . 4
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8 | 7 | anbi1i 431 |
. . 3
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9 | 1, 2, 8 | 3bitr4ri 202 |
. 2
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10 | elfz2 8651 |
. 2
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11 | eluz2 8255 |
. . 3
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12 | eluz2 8255 |
. . 3
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13 | 11, 12 | anbi12i 433 |
. 2
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14 | 9, 10, 13 | 3bitr4i 201 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-setind 4220 ax-cnex 6774 ax-resscn 6775 |
This theorem depends on definitions: df-bi 110 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-sbc 2759 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-neg 6982 df-z 8022 df-uz 8250 df-fz 8645 |
This theorem is referenced by: eluzfz 8655 elfzuz 8656 elfzuz3 8657 elfzuz2 8663 peano2fzr 8671 fzsplit2 8684 fzass4 8695 fzss1 8696 fzss2 8697 fzp1elp1 8707 fznn 8721 elfz2nn0 8743 elfzofz 8788 fzosplitsnm1 8835 fzofzp1b 8854 fzosplitsn 8859 iseqfveq2 8905 |
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