Theorem dfom3 9405

Description: The class of natural numbers ω can be defined as the intersection of all inductive sets (which is the smallest inductive set, since inductive sets are closed under intersection), which is valid provided we assume the Axiom of Infinity. Definition 6.3 of [Eisenberg] p. 82. (Contributed by NM, 6-Aug-1994.)

Assertion

Ref	Expression
dfom3	⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}

Distinct variable group:   𝑥,𝑦

Proof of Theorem dfom3

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 5231	. . . . 5 ⊢ ∅ ∈ V
2	1	elintab 4890	. . . 4 ⊢ (∅ ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → ∅ ∈ 𝑥))
3		simpl 483	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → ∅ ∈ 𝑥)
4	2, 3	mpgbir 1802	. . 3 ⊢ ∅ ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}
5		suceq 6331	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
6	5	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (suc 𝑦 ∈ 𝑥 ↔ suc 𝑧 ∈ 𝑥))
7	6	rspccv 3558	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥 → (𝑧 ∈ 𝑥 → suc 𝑧 ∈ 𝑥))
8	7	adantl 482	. . . . . . 7 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → (𝑧 ∈ 𝑥 → suc 𝑧 ∈ 𝑥))
9	8	a2i 14	. . . . . 6 ⊢ (((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) → ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → suc 𝑧 ∈ 𝑥))
10	9	alimi 1814	. . . . 5 ⊢ (∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) → ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → suc 𝑧 ∈ 𝑥))
11		vex 3436	. . . . . 6 ⊢ 𝑧 ∈ V
12	11	elintab 4890	. . . . 5 ⊢ (𝑧 ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
13	11	sucex 7656	. . . . . 6 ⊢ suc 𝑧 ∈ V
14	13	elintab 4890	. . . . 5 ⊢ (suc 𝑧 ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → suc 𝑧 ∈ 𝑥))
15	10, 12, 14	3imtr4i 292	. . . 4 ⊢ (𝑧 ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} → suc 𝑧 ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)})
16	15	rgenw 3076	. . 3 ⊢ ∀𝑧 ∈ ω (𝑧 ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} → suc 𝑧 ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)})
17		peano5 7740	. . 3 ⊢ ((∅ ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} ∧ ∀𝑧 ∈ ω (𝑧 ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} → suc 𝑧 ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)})) → ω ⊆ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)})
18	4, 16, 17	mp2an 689	. 2 ⊢ ω ⊆ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}
19		peano1 7735	. . . 4 ⊢ ∅ ∈ ω
20		peano2 7737	. . . . 5 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
21	20	rgen 3074	. . . 4 ⊢ ∀𝑦 ∈ ω suc 𝑦 ∈ ω
22		omex 9401	. . . . . 6 ⊢ ω ∈ V
23		eleq2 2827	. . . . . . . 8 ⊢ (𝑥 = ω → (∅ ∈ 𝑥 ↔ ∅ ∈ ω))
24		eleq2 2827	. . . . . . . . 9 ⊢ (𝑥 = ω → (suc 𝑦 ∈ 𝑥 ↔ suc 𝑦 ∈ ω))
25	24	raleqbi1dv 3340	. . . . . . . 8 ⊢ (𝑥 = ω → (∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥 ↔ ∀𝑦 ∈ ω suc 𝑦 ∈ ω))
26	23, 25	anbi12d 631	. . . . . . 7 ⊢ (𝑥 = ω → ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) ↔ (∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω)))
27		eleq2 2827	. . . . . . 7 ⊢ (𝑥 = ω → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ ω))
28	26, 27	imbi12d 345	. . . . . 6 ⊢ (𝑥 = ω → (((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ((∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω) → 𝑧 ∈ ω)))
29	22, 28	spcv 3544	. . . . 5 ⊢ (∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) → ((∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω) → 𝑧 ∈ ω))
30	12, 29	sylbi 216	. . . 4 ⊢ (𝑧 ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} → ((∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω) → 𝑧 ∈ ω))
31	19, 21, 30	mp2ani 695	. . 3 ⊢ (𝑧 ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} → 𝑧 ∈ ω)
32	31	ssriv 3925	. 2 ⊢ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} ⊆ ω
33	18, 32	eqssi 3937	1 ⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1537   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064   ⊆ wss 3887  ∅c0 4256  ∩ cint 4879  suc csuc 6268  ωcom 7712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-om 7713

This theorem is referenced by: (None)