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Theorem peano1 4213
Description: Zero is a natural number. One of Peano's 5 postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. (Contributed by NM, 15-May-1994.)
Assertion
Ref Expression
peano1 𝜔

Proof of Theorem peano1
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 3836 . . . 4 V
21elint 3573 . . 3 (∅ {y ∣ (∅ y x y suc x y)} ↔ z(z {y ∣ (∅ y x y suc x y)} → ∅ z))
3 df-clab 2009 . . . 4 (z {y ∣ (∅ y x y suc x y)} ↔ [z / y](∅ y x y suc x y))
4 ax-ia1 99 . . . . . 6 ((∅ y x y suc x y) → ∅ y)
54sbimi 1629 . . . . 5 ([z / y](∅ y x y suc x y) → [z / y]∅ y)
6 clelsb4 2125 . . . . 5 ([z / y]∅ y ↔ ∅ z)
75, 6sylib 127 . . . 4 ([z / y](∅ y x y suc x y) → ∅ z)
83, 7sylbi 114 . . 3 (z {y ∣ (∅ y x y suc x y)} → ∅ z)
92, 8mpgbir 1321 . 2 {y ∣ (∅ y x y suc x y)}
10 dfom3 4211 . 2 𝜔 = {y ∣ (∅ y x y suc x y)}
119, 10eleqtrri 2095 1 𝜔
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1375  [wsb 1627  {cab 2008  wral 2282  c0 3202   cint 3567  suc csuc 4026  𝜔com 4209
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-nul 3835
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2535  df-dif 2898  df-nul 3203  df-int 3568  df-iom 4210
This theorem is referenced by:  peano5  4217  limom  4232  frec0g  5868
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