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Mirrors > Home > NFE Home > Th. List > peano5 | Unicode version |
Description: The principle of mathematical induction: a set containing cardinal zero and closed under the successor operator is a superset of the finite cardinals. Theorem X.1.6 of [Rosser] p. 276. (Contributed by SF, 14-Jan-2015.) |
Ref | Expression |
---|---|
peano5 | 0c Nn 1c Nn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nncex 4396 | . . 3 Nn | |
2 | inexg 4100 | . . 3 Nn Nn | |
3 | 1, 2 | mpan 651 | . 2 Nn |
4 | peano1 4402 | . . 3 0c Nn | |
5 | elin 3219 | . . . 4 0c Nn 0c Nn 0c | |
6 | 5 | biimpri 197 | . . 3 0c Nn 0c 0c Nn |
7 | 4, 6 | mpan 651 | . 2 0c 0c Nn |
8 | elin 3219 | . . . . . 6 Nn Nn | |
9 | 8 | imbi1i 315 | . . . . 5 Nn 1c Nn 1c |
10 | impexp 433 | . . . . 5 Nn 1c Nn 1c | |
11 | 9, 10 | bitri 240 | . . . 4 Nn 1c Nn 1c |
12 | inss1 3475 | . . . . . . . 8 Nn Nn | |
13 | 12 | sseli 3269 | . . . . . . 7 Nn Nn |
14 | peano2 4403 | . . . . . . 7 Nn 1c Nn | |
15 | 13, 14 | syl 15 | . . . . . 6 Nn 1c Nn |
16 | elin 3219 | . . . . . . . 8 1c Nn 1c Nn 1c | |
17 | 16 | biimpri 197 | . . . . . . 7 1c Nn 1c 1c Nn |
18 | 17 | a1i 10 | . . . . . 6 Nn 1c Nn 1c 1c Nn |
19 | 15, 18 | mpand 656 | . . . . 5 Nn 1c 1c Nn |
20 | 19 | a2i 12 | . . . 4 Nn 1c Nn 1c Nn |
21 | 11, 20 | sylbir 204 | . . 3 Nn 1c Nn 1c Nn |
22 | 21 | ralimi2 2686 | . 2 Nn 1c Nn 1c Nn |
23 | df-nnc 4379 | . . . 4 Nn 0c 1c | |
24 | eleq2 2414 | . . . . . . . . 9 Nn 0c 0c Nn | |
25 | eleq2 2414 | . . . . . . . . . 10 Nn 1c 1c Nn | |
26 | 25 | raleqbi1dv 2815 | . . . . . . . . 9 Nn 1c Nn 1c Nn |
27 | 24, 26 | anbi12d 691 | . . . . . . . 8 Nn 0c 1c 0c Nn Nn 1c Nn |
28 | 27 | elabg 2986 | . . . . . . 7 Nn Nn 0c 1c 0c Nn Nn 1c Nn |
29 | 28 | biimprd 214 | . . . . . 6 Nn 0c Nn Nn 1c Nn Nn 0c 1c |
30 | 29 | 3impib 1149 | . . . . 5 Nn 0c Nn Nn 1c Nn Nn 0c 1c |
31 | intss1 3941 | . . . . 5 Nn 0c 1c 0c 1c Nn | |
32 | 30, 31 | syl 15 | . . . 4 Nn 0c Nn Nn 1c Nn 0c 1c Nn |
33 | 23, 32 | syl5eqss 3315 | . . 3 Nn 0c Nn Nn 1c Nn Nn Nn |
34 | inss2 3476 | . . 3 Nn | |
35 | 33, 34 | syl6ss 3284 | . 2 Nn 0c Nn Nn 1c Nn Nn |
36 | 3, 7, 22, 35 | syl3an 1224 | 1 0c Nn 1c Nn |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 358 w3a 934 wceq 1642 wcel 1710 cab 2339 wral 2614 cvv 2859 cin 3208 wss 3257 cint 3926 1cc1c 4134 Nn cnnc 4373 0cc0c 4374 cplc 4375 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-0c 4377 df-addc 4378 df-nnc 4379 |
This theorem is referenced by: findsd 4410 dmfrec 6314 |
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