Theorem peano1 4260

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 3875	. . . 4 ⊢ ∅ ∈ V
2	1	elint 3612	. . 3 ⊢ (∅ ∈ ∩ {y ∣ (∅ ∈ y ∧ ∀x ∈ y suc x ∈ y)} ↔ ∀z(z ∈ {y ∣ (∅ ∈ y ∧ ∀x ∈ y suc x ∈ y)} → ∅ ∈ z))
3		df-clab 2024	. . . 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 1644	. . . . 5 ⊢ ([z / y](∅ ∈ y ∧ ∀x ∈ y suc x ∈ y) → [z / y]∅ ∈ y)
6		clelsb4 2140	. . . . 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 1339	. 2 ⊢ ∅ ∈ ∩ {y ∣ (∅ ∈ y ∧ ∀x ∈ y suc x ∈ y)}
10		dfom3 4258	. 2 ⊢ 𝜔 = ∩ {y ∣ (∅ ∈ y ∧ ∀x ∈ y suc x ∈ y)}
11	9, 10	eleqtrri 2110	1 ⊢ ∅ ∈ 𝜔

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390  [wsb 1642  {cab 2023  ∀wral 2300  ∅c0 3218  ∩ cint 3606  suc csuc 4068  𝜔com 4256

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-nul 3219  df-int 3607  df-iom 4257

This theorem is referenced by:  peano5  4264  limom  4279  nnregexmid  4285  frec0g  5922  frecrdg  5931  oa1suc  5986  nna0r  5996  nnm0r  5997  nnmcl  5999  nnmsucr  6006  1onn  6029  nnm1  6033  nnaordex  6036  nnawordex  6037  0fin  6255  1lt2pi  6324  nq0m0r  6439  nq0a0  6440  prarloclem5  6483  frec2uzrand  8872  frecuzrdg0  8881  frecfzennn  8884  peano5setOLD  9400  bj-nn0suc  9424  bj-nn0sucALT  9438