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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4790 | . . . . 5 | |
2 | 1 | elintab 4487 | . . . 4 |
3 | simpl 473 | . . . 4 | |
4 | 2, 3 | mpgbir 1726 | . . 3 |
5 | suceq 5790 | . . . . . . . . . 10 | |
6 | 5 | eleq1d 2686 | . . . . . . . . 9 |
7 | 6 | rspccv 3306 | . . . . . . . 8 |
8 | 7 | adantl 482 | . . . . . . 7 |
9 | 8 | a2i 14 | . . . . . 6 |
10 | 9 | alimi 1739 | . . . . 5 |
11 | vex 3203 | . . . . . 6 | |
12 | 11 | elintab 4487 | . . . . 5 |
13 | 11 | sucex 7011 | . . . . . 6 |
14 | 13 | elintab 4487 | . . . . 5 |
15 | 10, 12, 14 | 3imtr4i 281 | . . . 4 |
16 | 15 | rgenw 2924 | . . 3 |
17 | peano5 7089 | . . 3 | |
18 | 4, 16, 17 | mp2an 708 | . 2 |
19 | peano1 7085 | . . . 4 | |
20 | peano2 7086 | . . . . 5 | |
21 | 20 | rgen 2922 | . . . 4 |
22 | omex 8540 | . . . . . 6 | |
23 | eleq2 2690 | . . . . . . . 8 | |
24 | eleq2 2690 | . . . . . . . . 9 | |
25 | 24 | raleqbi1dv 3146 | . . . . . . . 8 |
26 | 23, 25 | anbi12d 747 | . . . . . . 7 |
27 | eleq2 2690 | . . . . . . 7 | |
28 | 26, 27 | imbi12d 334 | . . . . . 6 |
29 | 22, 28 | spcv 3299 | . . . . 5 |
30 | 12, 29 | sylbi 207 | . . . 4 |
31 | 19, 21, 30 | mp2ani 714 | . . 3 |
32 | 31 | ssriv 3607 | . 2 |
33 | 18, 32 | eqssi 3619 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wal 1481 wceq 1483 wcel 1990 cab 2608 wral 2912 wss 3574 c0 3915 cint 4475 csuc 5725 com 7065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-om 7066 |
This theorem is referenced by: (None) |
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