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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 6973 | . 2 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω) | |
2 | 1 | biimpi 205 | 1 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 suc csuc 5642 ωcom 6957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-om 6958 |
This theorem is referenced by: onnseq 7328 seqomlem1 7432 seqomlem4 7435 onasuc 7495 onmsuc 7496 onesuc 7497 nnacl 7578 nnecl 7580 nnacom 7584 nnmsucr 7592 1onn 7606 2onn 7607 3onn 7608 4onn 7609 nnneo 7618 nneob 7619 omopthlem1 7622 onomeneq 8035 dif1en 8078 findcard 8084 findcard2 8085 unbnn2 8102 dffi3 8220 wofib 8333 axinf2 8420 dfom3 8427 noinfep 8440 cantnflt 8452 trcl 8487 cardsucnn 8694 dif1card 8716 fseqdom 8732 alephfp 8814 ackbij1lem16 8940 ackbij2lem2 8945 ackbij2lem3 8946 ackbij2 8948 sornom 8982 infpssrlem4 9011 fin23lem26 9030 fin23lem20 9042 fin23lem38 9054 fin23lem39 9055 isf32lem2 9059 isf32lem3 9060 isf34lem7 9084 isf34lem6 9085 fin1a2lem6 9110 fin1a2lem9 9113 fin1a2lem12 9116 domtriomlem 9147 axdc2lem 9153 axdc3lem 9155 axdc3lem2 9156 axdc3lem4 9158 axdc4lem 9160 axdclem2 9225 peano2nn 10909 om2uzrani 12613 uzrdgsuci 12621 fzennn 12629 axdc4uzlem 12644 bnj970 30271 trpredtr 30974 elhf2 31452 0hf 31454 hfsn 31456 hfpw 31462 neibastop2lem 31525 finxpsuclem 32410 |
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