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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 982 |
. . 3
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2 | an6 1332 |
. . 3
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3 | df-3an 982 |
. . . . 5
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4 | anandir 591 |
. . . . 5
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5 | ancom 266 |
. . . . . 6
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6 | 5 | anbi2i 457 |
. . . . 5
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7 | 3, 4, 6 | 3bitri 206 |
. . . 4
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8 | 7 | anbi1i 458 |
. . 3
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9 | 1, 2, 8 | 3bitr4ri 213 |
. 2
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10 | elfz2 10033 |
. 2
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11 | eluz2 9552 |
. . 3
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12 | eluz2 9552 |
. . 3
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13 | 11, 12 | anbi12i 460 |
. 2
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14 | 9, 10, 13 | 3bitr4i 212 |
1
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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 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-neg 8149 df-z 9272 df-uz 9547 df-fz 10027 |
This theorem is referenced by: eluzfz 10038 elfzuz 10039 elfzuz3 10040 elfzuz2 10047 peano2fzr 10055 fzsplit2 10068 fzass4 10080 fzss1 10081 fzss2 10082 fzp1elp1 10093 fznn 10107 elfz2nn0 10130 elfzofz 10180 fzosplitsnm1 10227 fzofzp1b 10246 fzosplitsn 10251 seq3fveq2 10487 monoord 10494 seq3id2 10527 bcn1 10756 seq3coll 10840 summodclem2a 11407 fisum0diag2 11473 mertenslemi1 11561 prodmodclem2a 11602 |
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