Theorem peano1 4317

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 3884	. . . 4 ⊢ ∅ ∈ V
2	1	elint 3621	. . 3 ⊢ (∅ ∈ ∩ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} → ∅ ∈ 𝑧))
3		df-clab 2027	. . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} ↔ [𝑧 / 𝑦](∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦))
4		simpl 102	. . . . . 6 ⊢ ((∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦) → ∅ ∈ 𝑦)
5	4	sbimi 1647	. . . . 5 ⊢ ([𝑧 / 𝑦](∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦) → [𝑧 / 𝑦]∅ ∈ 𝑦)
6		clelsb4 2143	. . . . 5 ⊢ ([𝑧 / 𝑦]∅ ∈ 𝑦 ↔ ∅ ∈ 𝑧)
7	5, 6	sylib 127	. . . 4 ⊢ ([𝑧 / 𝑦](∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦) → ∅ ∈ 𝑧)
8	3, 7	sylbi 114	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} → ∅ ∈ 𝑧)
9	2, 8	mpgbir 1342	. 2 ⊢ ∅ ∈ ∩ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)}
10		dfom3 4315	. 2 ⊢ ω = ∩ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)}
11	9, 10	eleqtrri 2113	1 ⊢ ∅ ∈ ω

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1393  [wsb 1645  {cab 2026  ∀wral 2306  ∅c0 3224  ∩ cint 3615  suc csuc 4102  ωcom 4313

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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-nul 3225  df-int 3616  df-iom 4314

This theorem is referenced by:  peano5  4321  limom  4336  nnregexmid  4342  frec0g  5983  frecrdg  5992  oa1suc  6047  nna0r  6057  nnm0r  6058  nnmcl  6060  nnmsucr  6067  1onn  6093  nnm1  6097  nnaordex  6100  nnawordex  6101  php5  6321  php5dom  6325  0fin  6341  findcard2  6346  findcard2s  6347  1lt2pi  6438  nq0m0r  6554  nq0a0  6555  prarloclem5  6598  frec2uzrand  9191  frecuzrdg0  9200  frecfzennn  9203  peano5setOLD  10065  bj-nn0suc  10089  bj-nn0sucALT  10103