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Mirrors > Home > ILE Home > Th. List > uzind4s | Unicode version |
Description: Induction on the upper
set of integers that starts at an integer ![]() |
Ref | Expression |
---|---|
uzind4s.1 |
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uzind4s.2 |
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Ref | Expression |
---|---|
uzind4s |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 2827 |
. 2
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2 | sbequ 1763 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | dfsbcq2 2827 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | dfsbcq2 2827 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | uzind4s.1 |
. 2
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6 | nfv 1462 |
. . . 4
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7 | nfs1v 1858 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | nfsbc1v 2842 |
. . . . 5
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9 | 7, 8 | nfim 1505 |
. . . 4
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10 | 6, 9 | nfim 1505 |
. . 3
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11 | eleq1 2145 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | sbequ12 1696 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | oveq1 5571 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 13 | sbceq1d 2829 |
. . . . 5
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15 | 12, 14 | imbi12d 232 |
. . . 4
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16 | 11, 15 | imbi12d 232 |
. . 3
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17 | uzind4s.2 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | 10, 16, 17 | chvar 1682 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 1, 2, 3, 4, 5, 18 | uzind4 8827 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3916 ax-pow 3968 ax-pr 3992 ax-un 4216 ax-setind 4308 ax-cnex 7199 ax-resscn 7200 ax-1cn 7201 ax-1re 7202 ax-icn 7203 ax-addcl 7204 ax-addrcl 7205 ax-mulcl 7206 ax-addcom 7208 ax-addass 7210 ax-distr 7212 ax-i2m1 7213 ax-0lt1 7214 ax-0id 7216 ax-rnegex 7217 ax-cnre 7219 ax-pre-ltirr 7220 ax-pre-ltwlin 7221 ax-pre-lttrn 7222 ax-pre-ltadd 7224 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2612 df-sbc 2825 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-int 3657 df-br 3806 df-opab 3860 df-mpt 3861 df-id 4076 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-rn 4402 df-res 4403 df-ima 4404 df-iota 4917 df-fun 4954 df-fn 4955 df-f 4956 df-fv 4960 df-riota 5520 df-ov 5567 df-oprab 5568 df-mpt2 5569 df-pnf 7287 df-mnf 7288 df-xr 7289 df-ltxr 7290 df-le 7291 df-sub 7418 df-neg 7419 df-inn 8177 df-n0 8426 df-z 8503 df-uz 8771 |
This theorem is referenced by: (None) |
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