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Theorem peano1 4611
Description: Zero is a natural number. One of Peano's five 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 4145 . . . 4 ∅ ∈ V
21elint 3865 . . 3 (∅ ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} → ∅ ∈ 𝑧))
3 df-clab 2176 . . . 4 (𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} ↔ [𝑧 / 𝑦](∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦))
4 simpl 109 . . . . . 6 ((∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦) → ∅ ∈ 𝑦)
54sbimi 1775 . . . . 5 ([𝑧 / 𝑦](∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦) → [𝑧 / 𝑦]∅ ∈ 𝑦)
6 clelsb2 2295 . . . . 5 ([𝑧 / 𝑦]∅ ∈ 𝑦 ↔ ∅ ∈ 𝑧)
75, 6sylib 122 . . . 4 ([𝑧 / 𝑦](∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦) → ∅ ∈ 𝑧)
83, 7sylbi 121 . . 3 (𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} → ∅ ∈ 𝑧)
92, 8mpgbir 1464 . 2 ∅ ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}
10 dfom3 4609 . 2 ω = {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}
119, 10eleqtrri 2265 1 ∅ ∈ ω
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  [wsb 1773  wcel 2160  {cab 2175  wral 2468  c0 3437   cint 3859  suc csuc 4383  ωcom 4607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-nul 4144
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-nul 3438  df-int 3860  df-iom 4608
This theorem is referenced by:  peano5  4615  limom  4631  nnregexmid  4638  omsinds  4639  nnpredcl  4640  frec0g  6422  frecabcl  6424  frecrdg  6433  oa1suc  6492  nna0r  6503  nnm0r  6504  nnmcl  6506  nnmsucr  6513  1onn  6545  nnm1  6550  nnaordex  6553  nnawordex  6554  php5  6886  php5dom  6891  0fin  6912  findcard2  6917  findcard2s  6918  infm  6932  inffiexmid  6934  0ct  7136  ctmlemr  7137  ctssdclemn0  7139  ctssdc  7142  omct  7146  nninfisol  7161  fodjum  7174  fodju0  7175  ctssexmid  7178  nninfwlpoimlemg  7203  nninfwlpoimlemginf  7204  1lt2pi  7369  nq0m0r  7485  nq0a0  7486  prarloclem5  7529  frec2uzrand  10436  frecuzrdg0  10444  frecuzrdg0t  10453  frecfzennn  10457  0tonninf  10470  1tonninf  10471  hashinfom  10790  hashunlem  10816  hash1  10823  ennnfonelemj0  12452  ennnfonelem1  12458  ennnfonelemhf1o  12464  ennnfonelemhom  12466  fnpr2o  12815  fvpr0o  12817  xpscf  12823  bj-nn0suc  15174  bj-nn0sucALT  15188  012of  15204  2o01f  15205  pwle2  15207  pwf1oexmid  15208  subctctexmid  15209  peano3nninf  15215  nninfall  15217  nninfsellemdc  15218  nninfsellemeq  15222  nninffeq  15228  isomninnlem  15237  iswomninnlem  15256  ismkvnnlem  15259
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