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Mirrors > Home > MPE Home > Th. List > dfom3 | Structured version Visualization version Unicode version |
Description: The class of natural numbers omega can be defined as the smallest "inductive set," which is valid provided we assume the Axiom of Infinity. Definition 6.3 of [Eisenberg] p. 82. (Contributed by NM, 6-Aug-1994.) |
Ref | Expression |
---|---|
dfom3 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4548 |
. . . . 5
![]() ![]() ![]() ![]() | |
2 | 1 | elintab 4258 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | simpl 463 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | 2, 3 | mpgbir 1683 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | suceq 5506 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 5 | eleq1d 2523 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | 6 | rspccv 3158 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | 7 | adantl 472 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | 8 | a2i 14 |
. . . . . 6
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10 | 9 | alimi 1694 |
. . . . 5
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11 | vex 3059 |
. . . . . 6
![]() ![]() ![]() ![]() | |
12 | 11 | elintab 4258 |
. . . . 5
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13 | 11 | sucex 6664 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() |
14 | 13 | elintab 4258 |
. . . . 5
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15 | 10, 12, 14 | 3imtr4i 274 |
. . . 4
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16 | 15 | rgenw 2760 |
. . 3
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17 | peano5 6742 |
. . 3
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18 | 4, 16, 17 | mp2an 683 |
. 2
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19 | peano1 6738 |
. . . 4
![]() ![]() ![]() ![]() | |
20 | peano2 6739 |
. . . . 5
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21 | 20 | rgen 2758 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | omex 8173 |
. . . . . 6
![]() ![]() ![]() ![]() | |
23 | eleq2 2528 |
. . . . . . . 8
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24 | eleq2 2528 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 24 | raleqbi1dv 3006 |
. . . . . . . 8
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26 | 23, 25 | anbi12d 722 |
. . . . . . 7
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27 | eleq2 2528 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 26, 27 | imbi12d 326 |
. . . . . 6
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29 | 22, 28 | spcv 3151 |
. . . . 5
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30 | 12, 29 | sylbi 200 |
. . . 4
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31 | 19, 21, 30 | mp2ani 689 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 31 | ssriv 3447 |
. 2
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33 | 18, 32 | eqssi 3459 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-8 1899 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-sep 4538 ax-nul 4547 ax-pr 4652 ax-un 6609 ax-inf2 8171 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-eu 2313 df-mo 2314 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-ral 2753 df-rex 2754 df-rab 2757 df-v 3058 df-sbc 3279 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-pss 3431 df-nul 3743 df-if 3893 df-pw 3964 df-sn 3980 df-pr 3982 df-tp 3984 df-op 3986 df-uni 4212 df-int 4248 df-br 4416 df-opab 4475 df-tr 4511 df-eprel 4763 df-po 4773 df-so 4774 df-fr 4811 df-we 4813 df-ord 5444 df-on 5445 df-lim 5446 df-suc 5447 df-om 6719 |
This theorem is referenced by: (None) |
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