Theorem peano1 4736

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.

Step	Hyp	Ref	Expression
1		0ex 4255	. . . 4 ⊢ ∅ ∈ V
2	1	elint 3971	. . 3 ⊢ (∅ ∈ ∩ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} → ∅ ∈ 𝑧))
3		df-clab 2225	. . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} ↔ [𝑧 / 𝑦](∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦))
4		simpl 109	. . . . . 6 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦) → ∅ ∈ 𝑦)
5	4	sbimi 1817	. . . . 5 ⊢ ([𝑧 / 𝑦](∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦) → [𝑧 / 𝑦]∅ ∈ 𝑦)
6		clelsb2 2344	. . . . 5 ⊢ ([𝑧 / 𝑦]∅ ∈ 𝑦 ↔ ∅ ∈ 𝑧)
7	5, 6	sylib 122	. . . 4 ⊢ ([𝑧 / 𝑦](∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦) → ∅ ∈ 𝑧)
8	3, 7	sylbi 121	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} → ∅ ∈ 𝑧)
9	2, 8	mpgbir 1506	. 2 ⊢ ∅ ∈ ∩ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)}
10		dfom3 4734	. 2 ⊢ ω = ∩ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)}
11	9, 10	eleqtrri 2314	1 ⊢ ∅ ∈ ω

Syntax hints:   → wi 4   ∧ wa 104  [wsb 1815   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∅c0 3520  ∩ cint 3965  suc csuc 4505  ωcom 4732

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-nul 4254

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-nul 3521  df-int 3966  df-iom 4733

This theorem is referenced by:  peano5  4740  limom  4756  nnregexmid  4763  omsinds  4764  nnpredcl  4765  frec0g  6658  frecabcl  6660  frecrdg  6669  oa1suc  6730  nna0r  6741  nnm0r  6742  nnmcl  6744  nnmsucr  6751  1onn  6783  nnm1  6788  nnaordex  6791  nnawordex  6792  php5  7149  php5dom  7154  0fi  7178  findcard2  7183  findcard2s  7184  infm  7201  inffiexmid  7203  0ct  7437  ctmlemr  7438  ctssdclemn0  7440  ctssdc  7443  omct  7447  nninfisol  7463  fodjum  7476  fodju0  7477  ctssexmid  7480  nninfwlpoimlemg  7505  nninfwlpoimlemginf  7506  1lt2pi  7697  nq0m0r  7813  nq0a0  7814  prarloclem5  7857  frec2uzrand  10820  frecuzrdg0  10828  frecuzrdg0t  10837  frecfzennn  10841  0tonninf  10855  1tonninf  10856  hashinfom  11195  hashunlem  11222  hash1  11230  nninfctlemfo  12795  ennnfonelemj0  13270  ennnfonelem1  13276  ennnfonelemhf1o  13282  ennnfonelemhom  13284  fnpr2o  13637  fvpr0o  13639  xpscf  13645  bj-nn0suc  16904  bj-nn0sucALT  16918  012of  16937  2o01f  16938  pwle2  16942  pwf1oexmid  16943  subctctexmid  16944  peano3nninf  16955  nninfall  16957  nninfsellemdc  16958  nninfsellemeq  16962  nninffeq  16968  nnnninfex  16970  isomninnlem  16984  iswomninnlem  17004  ismkvnnlem  17007