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Mirrors > Home > MPE Home > Th. List > peano2 | Structured version Visualization version GIF version |
Description: The successor of any natural number is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(2) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.) |
Ref | Expression |
---|---|
peano2 | ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2b 7359 | . 2 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω) | |
2 | 1 | biimpi 208 | 1 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 suc csuc 5978 ωcom 7343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-tr 4988 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-om 7344 |
This theorem is referenced by: onnseq 7724 seqomlem1 7828 seqomlem4 7831 onasuc 7892 onmsuc 7893 onesuc 7894 nnacl 7975 nnecl 7977 nnacom 7981 nnmsucr 7989 1onn 8003 2onn 8004 3onn 8005 4onn 8006 nnneo 8015 nneob 8016 omopthlem1 8019 onomeneq 8438 dif1en 8481 findcard 8487 findcard2 8488 unbnn2 8505 dffi3 8625 wofib 8739 axinf2 8834 dfom3 8841 noinfep 8854 cantnflt 8866 trcl 8901 cardsucnn 9144 dif1card 9166 fseqdom 9182 alephfp 9264 ackbij1lem5 9381 ackbij1lem16 9392 ackbij2lem2 9397 ackbij2lem3 9398 ackbij2 9400 sornom 9434 infpssrlem4 9463 fin23lem26 9482 fin23lem20 9494 fin23lem38 9506 fin23lem39 9507 isf32lem2 9511 isf32lem3 9512 isf34lem7 9536 isf34lem6 9537 fin1a2lem6 9562 fin1a2lem9 9565 fin1a2lem12 9568 domtriomlem 9599 axdc2lem 9605 axdc3lem 9607 axdc3lem2 9608 axdc3lem4 9610 axdc4lem 9612 axdclem2 9677 peano2nn 11388 om2uzrani 13070 uzrdgsuci 13078 fzennn 13086 axdc4uzlem 13101 bnj970 31616 trpredtr 32318 elhf2 32871 0hf 32873 hfsn 32875 hfpw 32881 neibastop2lem 32943 finxpsuclem 33829 |
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