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Theorem dfnn2 11225
Description: Alternate definition of the set of positive integers. This was our original definition, before the current df-nn 11213 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 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
Distinct variable group:   𝑥,𝑦

Proof of Theorem dfnn2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 1ex 10227 . . . . 5 1 ∈ V
21elintab 4639 . . . 4 (1 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → 1 ∈ 𝑥))
3 simpl 474 . . . 4 ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → 1 ∈ 𝑥)
42, 3mpgbir 1875 . . 3 1 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
5 oveq1 6820 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑦 + 1) = (𝑧 + 1))
65eleq1d 2824 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑧 + 1) ∈ 𝑥))
76rspccv 3446 . . . . . . . 8 (∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥 → (𝑧𝑥 → (𝑧 + 1) ∈ 𝑥))
87adantl 473 . . . . . . 7 ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → (𝑧𝑥 → (𝑧 + 1) ∈ 𝑥))
98a2i 14 . . . . . 6 (((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → 𝑧𝑥) → ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → (𝑧 + 1) ∈ 𝑥))
109alimi 1888 . . . . 5 (∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → 𝑧𝑥) → ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → (𝑧 + 1) ∈ 𝑥))
11 vex 3343 . . . . . 6 𝑧 ∈ V
1211elintab 4639 . . . . 5 (𝑧 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → 𝑧𝑥))
13 ovex 6841 . . . . . 6 (𝑧 + 1) ∈ V
1413elintab 4639 . . . . 5 ((𝑧 + 1) ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → (𝑧 + 1) ∈ 𝑥))
1510, 12, 143imtr4i 281 . . . 4 (𝑧 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} → (𝑧 + 1) ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)})
1615rgen 3060 . . 3 𝑧 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} (𝑧 + 1) ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
17 peano5nni 11215 . . 3 ((1 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ∧ ∀𝑧 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} (𝑧 + 1) ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}) → ℕ ⊆ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)})
184, 16, 17mp2an 710 . 2 ℕ ⊆ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
19 1nn 11223 . . . 4 1 ∈ ℕ
20 peano2nn 11224 . . . . 5 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
2120rgen 3060 . . . 4 𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ
22 nnex 11218 . . . . 5 ℕ ∈ V
23 eleq2 2828 . . . . . 6 (𝑥 = ℕ → (1 ∈ 𝑥 ↔ 1 ∈ ℕ))
24 eleq2 2828 . . . . . . 7 (𝑥 = ℕ → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ ℕ))
2524raleqbi1dv 3285 . . . . . 6 (𝑥 = ℕ → (∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ))
2623, 25anbi12d 749 . . . . 5 (𝑥 = ℕ → ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ)))
2722, 26elab 3490 . . . 4 (ℕ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ))
2819, 21, 27mpbir2an 993 . . 3 ℕ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
29 intss1 4644 . . 3 (ℕ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} → {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ⊆ ℕ)
3028, 29ax-mp 5 . 2 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ⊆ ℕ
3118, 30eqssi 3760 1 ℕ = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1630   = wceq 1632  wcel 2139  {cab 2746  wral 3050  wss 3715   cint 4627  (class class class)co 6813  1c1 10129   + caddc 10131  cn 11212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-i2m1 10196  ax-1ne0 10197  ax-rrecex 10200  ax-cnre 10201
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-nn 11213
This theorem is referenced by:  dfnn3  11226
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