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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 929 |
. . 3
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2 | an6 1264 |
. . 3
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3 | df-3an 929 |
. . . . 5
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4 | anandir 559 |
. . . . 5
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5 | ancom 263 |
. . . . . 6
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6 | 5 | anbi2i 446 |
. . . . 5
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7 | 3, 4, 6 | 3bitri 205 |
. . . 4
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8 | 7 | anbi1i 447 |
. . 3
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9 | 1, 2, 8 | 3bitr4ri 212 |
. 2
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10 | elfz2 9580 |
. 2
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11 | eluz2 9124 |
. . 3
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12 | eluz2 9124 |
. . 3
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13 | 11, 12 | anbi12i 449 |
. 2
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14 | 9, 10, 13 | 3bitr4i 211 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 |
This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-fv 5057 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-neg 7753 df-z 8849 df-uz 9119 df-fz 9574 |
This theorem is referenced by: eluzfz 9584 elfzuz 9585 elfzuz3 9586 elfzuz2 9592 peano2fzr 9600 fzsplit2 9613 fzass4 9625 fzss1 9626 fzss2 9627 fzp1elp1 9638 fznn 9652 elfz2nn0 9675 elfzofz 9722 fzosplitsnm1 9769 fzofzp1b 9788 fzosplitsn 9793 seq3fveq2 10019 monoord 10026 seq3id2 10059 bcn1 10281 seq3coll 10362 summodclem2a 10924 fisum0diag2 10990 mertenslemi1 11078 |
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