Theorem dfom3 9576

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 5257	. . . . 5 ⊢ ∅ ∈ V
2	1	elintab 4918	. . . 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 6388	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
6	5	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (suc 𝑦 ∈ 𝑥 ↔ suc 𝑧 ∈ 𝑥))
7	6	rspccv 3582	. . . . . . . 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 3448	. . . . . 6 ⊢ 𝑧 ∈ V
12	11	elintab 4918	. . . . 5 ⊢ (𝑧 ∈ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑥))
13	11	sucex 7762	. . . . . 6 ⊢ suc 𝑧 ∈ V
14	13	elintab 4918	. . . . 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 7849	. . 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 7845	. . . 4 ⊢ ∅ ∈ ω
20		peano2 7846	. . . . 5 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
21	20	rgen 3046	. . . 4 ⊢ ∀𝑦 ∈ ω suc 𝑦 ∈ ω
22		omex 9572	. . . . . 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 3308	. . . . . . . 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 3568	. . . . 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 3947	. 2 ⊢ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} ⊆ ω
33	18, 32	eqssi 3960	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 3911  ∅c0 4292  ∩ cint 4906  suc csuc 6322  ωcom 7822

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 5246  ax-nul 5256  ax-pr 5382  ax-un 7691  ax-inf2 9570

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 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-br 5103  df-opab 5165  df-tr 5210  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-om 7823

This theorem is referenced by: (None)