Theorem dfom3 9600

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. Definition 1.20 of [Schloeder] p. 3. (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 5262	. . . . 5 ⊢ ∅ ∈ V
2	1	elintab 4922	. . . 4 ⊢ (∅ ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → ∅ ∈ 𝑥))
3		simpl 482	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → ∅ ∈ 𝑥)
4	2, 3	mpgbir 1799	. . 3 ⊢ ∅ ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}
5		suceq 6400	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
6	5	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (suc 𝑦 ∈ 𝑥 ↔ suc 𝑧 ∈ 𝑥))
7	6	rspccv 3585	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥 → (𝑧 ∈ 𝑥 → suc 𝑧 ∈ 𝑥))
8	7	adantl 481	. . . . . . 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 1811	. . . . 5 ⊢ (∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) → ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → suc 𝑧 ∈ 𝑥))
11		vex 3451	. . . . . 6 ⊢ 𝑧 ∈ V
12	11	elintab 4922	. . . . 5 ⊢ (𝑧 ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
13	11	sucex 7782	. . . . . 6 ⊢ suc 𝑧 ∈ V
14	13	elintab 4922	. . . . 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 3048	. . 3 ⊢ ∀𝑧 ∈ ω (𝑧 ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} → suc 𝑧 ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)})
17		peano5 7869	. . 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 692	. 2 ⊢ ω ⊆ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}
19		peano1 7865	. . . 4 ⊢ ∅ ∈ ω
20		peano2 7866	. . . . 5 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
21	20	rgen 3046	. . . 4 ⊢ ∀𝑦 ∈ ω suc 𝑦 ∈ ω
22		omex 9596	. . . . . 6 ⊢ ω ∈ V
23		eleq2 2817	. . . . . . . 8 ⊢ (𝑥 = ω → (∅ ∈ 𝑥 ↔ ∅ ∈ ω))
24		eleq2 2817	. . . . . . . . 9 ⊢ (𝑥 = ω → (suc 𝑦 ∈ 𝑥 ↔ suc 𝑦 ∈ ω))
25	24	raleqbi1dv 3311	. . . . . . . 8 ⊢ (𝑥 = ω → (∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥 ↔ ∀𝑦 ∈ ω suc 𝑦 ∈ ω))
26	23, 25	anbi12d 632	. . . . . . 7 ⊢ (𝑥 = ω → ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) ↔ (∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω)))
27		eleq2 2817	. . . . . . 7 ⊢ (𝑥 = ω → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ ω))
28	26, 27	imbi12d 344	. . . . . 6 ⊢ (𝑥 = ω → (((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) ↔ ((∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω) → 𝑧 ∈ ω)))
29	22, 28	spcv 3571	. . . . 5 ⊢ (∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥) → ((∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω) → 𝑧 ∈ ω))
30	12, 29	sylbi 217	. . . 4 ⊢ (𝑧 ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} → ((∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω) → 𝑧 ∈ ω))
31	19, 21, 30	mp2ani 698	. . 3 ⊢ (𝑧 ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} → 𝑧 ∈ ω)
32	31	ssriv 3950	. 2 ⊢ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} ⊆ ω
33	18, 32	eqssi 3963	1 ⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044   ⊆ wss 3914  ∅c0 4296  ∩ cint 4910  suc csuc 6334  ωcom 7842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711  ax-inf2 9594

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-br 5108  df-opab 5170  df-tr 5215  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-om 7843

This theorem is referenced by: (None)