Theorem dfom3 9666

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 5282	. . . . 5 ⊢ ∅ ∈ V
2	1	elintab 4939	. . . 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 6424	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
6	5	eleq1d 2820	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (suc 𝑦 ∈ 𝑥 ↔ suc 𝑧 ∈ 𝑥))
7	6	rspccv 3603	. . . . . . . 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 3468	. . . . . 6 ⊢ 𝑧 ∈ V
12	11	elintab 4939	. . . . 5 ⊢ (𝑧 ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
13	11	sucex 7805	. . . . . 6 ⊢ suc 𝑧 ∈ V
14	13	elintab 4939	. . . . 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 3056	. . 3 ⊢ ∀𝑧 ∈ ω (𝑧 ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} → suc 𝑧 ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)})
17		peano5 7894	. . 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 7889	. . . 4 ⊢ ∅ ∈ ω
20		peano2 7891	. . . . 5 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
21	20	rgen 3054	. . . 4 ⊢ ∀𝑦 ∈ ω suc 𝑦 ∈ ω
22		omex 9662	. . . . . 6 ⊢ ω ∈ V
23		eleq2 2824	. . . . . . . 8 ⊢ (𝑥 = ω → (∅ ∈ 𝑥 ↔ ∅ ∈ ω))
24		eleq2 2824	. . . . . . . . 9 ⊢ (𝑥 = ω → (suc 𝑦 ∈ 𝑥 ↔ suc 𝑦 ∈ ω))
25	24	raleqbi1dv 3321	. . . . . . . 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 2824	. . . . . . 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 3589	. . . . 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 3967	. 2 ⊢ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} ⊆ ω
33	18, 32	eqssi 3980	1 ⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109  {cab 2714  ∀wral 3052   ⊆ wss 3931  ∅c0 4313  ∩ cint 4927  suc csuc 6359  ωcom 7866

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 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734  ax-inf2 9660

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 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-br 5125  df-opab 5187  df-tr 5235  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-om 7867

This theorem is referenced by: (None)