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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 7904 | . 2 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω) | |
| 2 | 1 | biimpi 216 | 1 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 suc csuc 6386 ωcom 7887 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-om 7888 |
| This theorem is referenced by: onnseq 8384 seqomlem1 8490 seqomlem4 8493 onasuc 8566 onmsuc 8567 onesuc 8568 o2p2e4 8579 nnacl 8649 nnecl 8651 nnacom 8655 nnmsucr 8663 nnaordex2 8677 1onnALT 8679 2onnALT 8681 3onn 8682 4onn 8683 nnneo 8693 nneob 8694 omopthlem1 8697 eldifsucnn 8702 findcard 9203 unfi 9211 phplem1 9244 php 9247 onomeneqOLD 9266 dif1ennnALT 9311 unbnn2 9333 dffi3 9471 wofib 9585 axinf2 9680 dfom3 9687 noinfep 9700 cantnflt 9712 ttrcltr 9756 ttrclss 9760 ttrclselem2 9766 trcl 9768 cardsucnn 10025 harsucnn 10038 dif1card 10050 fseqdom 10066 alephfp 10148 ackbij1lem5 10263 ackbij1lem16 10274 ackbij2lem2 10279 ackbij2lem3 10280 ackbij2 10282 sornom 10317 infpssrlem4 10346 fin23lem26 10365 fin23lem20 10377 fin23lem38 10389 fin23lem39 10390 isf32lem2 10394 isf32lem3 10395 isf34lem7 10419 isf34lem6 10420 fin1a2lem6 10445 fin1a2lem9 10448 fin1a2lem12 10451 domtriomlem 10482 axdc2lem 10488 axdc3lem 10490 axdc3lem2 10491 axdc3lem4 10493 axdc4lem 10495 axdclem2 10560 peano2nn 12278 om2uzrani 13993 uzrdgsuci 14001 fzennn 14009 axdc4uzlem 14024 precsexlem4 28234 precsexlem5 28235 precsexlem11 28241 noseqp1 28297 om2noseqlt 28305 noseqrdgsuc 28314 n0sbday 28354 dfnns2 28362 pw2bday 28418 zs12bday 28424 constrextdg2lem 33789 bnj970 34961 satfvsuc 35366 satfvsucsuc 35370 gonarlem 35399 goalrlem 35401 satffunlem2lem2 35411 satffunlem2 35413 ex-sategoelelomsuc 35431 elhf2 36176 0hf 36178 hfsn 36180 hfpw 36186 neibastop2lem 36361 exrecfnlem 37380 finxpsuclem 37398 domalom 37405 onexoegt 43256 nnoeomeqom 43325 nna1iscard 43558 omssaxinf2 45005 |
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