Theorem dfnn2 12166

Description: Alternate definition of the set of positive integers. This was our original definition, before the current df-nn 12154 replaced it. This definition requires the axiom of infinity to ensure it has the properties we expect. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)

Assertion

Ref	Expression
dfnn2	⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Distinct variable group:   𝑥,𝑦

Proof of Theorem dfnn2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1ex 11151	. . . . 5 ⊢ 1 ∈ V
2	1	elintab 4919	. . . 4 ⊢ (1 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) → 1 ∈ 𝑥))
3		simpl 483	. . . 4 ⊢ ((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) → 1 ∈ 𝑥)
4	2, 3	mpgbir 1801	. . 3 ⊢ 1 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}
5		oveq1 7364	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑦 + 1) = (𝑧 + 1))
6	5	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑧 + 1) ∈ 𝑥))
7	6	rspccv 3578	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥 → (𝑧 ∈ 𝑥 → (𝑧 + 1) ∈ 𝑥))
8	7	adantl 482	. . . . . . 7 ⊢ ((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) → (𝑧 ∈ 𝑥 → (𝑧 + 1) ∈ 𝑥))
9	8	a2i 14	. . . . . 6 ⊢ (((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) → 𝑧 ∈ 𝑥) → ((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) → (𝑧 + 1) ∈ 𝑥))
10	9	alimi 1813	. . . . 5 ⊢ (∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) → 𝑧 ∈ 𝑥) → ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) → (𝑧 + 1) ∈ 𝑥))
11		vex 3449	. . . . . 6 ⊢ 𝑧 ∈ V
12	11	elintab 4919	. . . . 5 ⊢ (𝑧 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) → 𝑧 ∈ 𝑥))
13		ovex 7390	. . . . . 6 ⊢ (𝑧 + 1) ∈ V
14	13	elintab 4919	. . . . 5 ⊢ ((𝑧 + 1) ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) → (𝑧 + 1) ∈ 𝑥))
15	10, 12, 14	3imtr4i 291	. . . 4 ⊢ (𝑧 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} → (𝑧 + 1) ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)})
16	15	rgen 3066	. . 3 ⊢ ∀𝑧 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} (𝑧 + 1) ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}
17		peano5nni 12156	. . 3 ⊢ ((1 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ∧ ∀𝑧 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} (𝑧 + 1) ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}) → ℕ ⊆ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)})
18	4, 16, 17	mp2an 690	. 2 ⊢ ℕ ⊆ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}
19		1nn 12164	. . . 4 ⊢ 1 ∈ ℕ
20		peano2nn 12165	. . . . 5 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
21	20	rgen 3066	. . . 4 ⊢ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ
22		nnex 12159	. . . . 5 ⊢ ℕ ∈ V
23		eleq2 2826	. . . . . 6 ⊢ (𝑥 = ℕ → (1 ∈ 𝑥 ↔ 1 ∈ ℕ))
24		eleq2 2826	. . . . . . 7 ⊢ (𝑥 = ℕ → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ ℕ))
25	24	raleqbi1dv 3307	. . . . . 6 ⊢ (𝑥 = ℕ → (∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ))
26	23, 25	anbi12d 631	. . . . 5 ⊢ (𝑥 = ℕ → ((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ)))
27	22, 26	elab 3630	. . . 4 ⊢ (ℕ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ))
28	19, 21, 27	mpbir2an 709	. . 3 ⊢ ℕ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}
29		intss1 4924	. . 3 ⊢ (ℕ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} → ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ⊆ ℕ)
30	28, 29	ax-mp 5	. 2 ⊢ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ⊆ ℕ
31	18, 30	eqssi 3960	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1539   = wceq 1541   ∈ wcel 2106  {cab 2713  ∀wral 3064   ⊆ wss 3910  ∩ cint 4907  (class class class)co 7357  1c1 11052   + caddc 11054  ℕcn 12153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-1cn 11109  ax-addcl 11111

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-nn 12154

This theorem is referenced by:  dfnn3  12167