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Mirrors > Home > MPE Home > Th. List > peano1 | Structured version Visualization version GIF version |
Description: Zero is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. Note: Unlike most textbooks, our proofs of peano1 7900 through peano5 7905 do not use the Axiom of Infinity. Unlike Takeuti and Zaring, they also do not use the Axiom of Regularity. (Contributed by NM, 15-May-1994.) Avoid ax-un 7746. (Revised by BTernaryTau, 29-Nov-2024.) |
Ref | Expression |
---|---|
peano1 | ⊢ ∅ ∈ ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elon 6428 | . 2 ⊢ ∅ ∈ On | |
2 | 0ellim 6437 | . . 3 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥) | |
3 | 2 | ax-gen 1789 | . 2 ⊢ ∀𝑥(Lim 𝑥 → ∅ ∈ 𝑥) |
4 | elom 7879 | . 2 ⊢ (∅ ∈ ω ↔ (∅ ∈ On ∧ ∀𝑥(Lim 𝑥 → ∅ ∈ 𝑥))) | |
5 | 1, 3, 4 | mpbir2an 709 | 1 ⊢ ∅ ∈ ω |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 ∈ wcel 2098 ∅c0 4326 Oncon0 6374 Lim wlim 6375 ωcom 7876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-tr 5270 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-ord 6377 df-on 6378 df-lim 6379 df-om 7877 |
This theorem is referenced by: onnseq 8371 rdg0 8448 fr0g 8463 seqomlem3 8479 oa1suc 8558 o2p2e4 8568 om1 8569 oe1 8571 nna0r 8636 nnm0r 8637 nnmcl 8639 nnecl 8640 nnmsucr 8652 nnaword1 8656 nnaordex 8665 1onnALT 8668 oaabs2 8676 nnm1 8679 nneob 8683 omopth 8689 snfi 9075 0fin 9202 0sdom1domALT 9270 isinf 9291 findcard2OLD 9315 nnunifi 9325 unblem2 9327 infn0 9338 infn0ALT 9339 unfilem3 9343 dffi3 9462 inf0 9652 infeq5i 9667 axinf2 9671 dfom3 9678 infdifsn 9688 noinfep 9691 cantnflt 9703 cnfcomlem 9730 cnfcom 9731 cnfcom2lem 9732 cnfcom3lem 9734 cnfcom3 9735 brttrcl2 9745 ttrcltr 9747 rnttrcl 9753 trcl 9759 rankdmr1 9832 rankeq0b 9891 cardlim 10003 infxpenc 10049 infxpenc2 10053 alephgeom 10113 alephfplem4 10138 ackbij1lem13 10263 ackbij1 10269 ackbij1b 10270 ominf4 10343 fin23lem16 10366 fin23lem31 10374 fin23lem40 10382 isf32lem9 10392 isf34lem7 10410 isf34lem6 10411 fin1a2lem6 10436 fin1a2lem7 10437 fin1a2lem11 10441 axdc3lem2 10482 axdc3lem4 10484 axdc4lem 10486 axcclem 10488 axdclem2 10551 pwfseqlem5 10694 omina 10722 wunex3 10772 1lt2pi 10936 1nn 12261 om2uzrani 13957 uzrdg0i 13964 fzennn 13973 axdc4uzlem 13988 hash1 14403 fnpr2o 17546 fvpr0o 17548 ltbwe 21989 2ndcdisj2 23381 precsexlem11 28135 noseq0 28183 noseqrdg0 28200 n0sbday 28237 snct 32516 goelel3xp 34991 satfv0 35001 satfv1 35006 satf0 35015 satf00 35017 satf0suclem 35018 sat1el2xp 35022 fmla0 35025 fmlasuc0 35027 fmla1 35030 gonan0 35035 gonar 35038 goalr 35040 satffunlem1lem2 35046 satffunlem1 35050 satefvfmla0 35061 prv0 35073 nnuni 35354 0hf 35806 neibastop2lem 35877 bj-rdg0gALT 36583 rdgeqoa 36882 exrecfnlem 36891 finxp0 36903 onexomgt 42700 onexoegt 42703 omnord1 42765 oenord1 42776 oaomoencom 42777 cantnftermord 42780 cantnfub 42781 cantnf2 42785 dflim5 42789 oacl2g 42790 onmcl 42791 omabs2 42792 omcl2 42793 tfsconcat0b 42806 ofoaf 42815 ofoafo 42816 ofoaid1 42818 ofoaid2 42819 naddcnff 42822 naddcnffo 42824 naddcnfid1 42827 naddcnfid2 42828 0finon 42909 0iscard 43002 |
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