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Theorem dfnn2 11289
Description: Alternate definition of the set of positive integers. This was our original definition, before the current df-nn 11275 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 10289 . . . . 5 1 ∈ V
21elintab 4644 . . . 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 1894 . . 3 1 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
5 oveq1 6849 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑦 + 1) = (𝑧 + 1))
65eleq1d 2829 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑧 + 1) ∈ 𝑥))
76rspccv 3458 . . . . . . . 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 1906 . . . . 5 (∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → 𝑧𝑥) → ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → (𝑧 + 1) ∈ 𝑥))
11 vex 3353 . . . . . 6 𝑧 ∈ V
1211elintab 4644 . . . . 5 (𝑧 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → 𝑧𝑥))
13 ovex 6874 . . . . . 6 (𝑧 + 1) ∈ V
1413elintab 4644 . . . . 5 ((𝑧 + 1) ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → (𝑧 + 1) ∈ 𝑥))
1510, 12, 143imtr4i 283 . . . 4 (𝑧 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} → (𝑧 + 1) ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)})
1615rgen 3069 . . 3 𝑧 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} (𝑧 + 1) ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
17 peano5nni 11277 . . 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 683 . 2 ℕ ⊆ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
19 1nn 11287 . . . 4 1 ∈ ℕ
20 peano2nn 11288 . . . . 5 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
2120rgen 3069 . . . 4 𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ
22 nnex 11281 . . . . 5 ℕ ∈ V
23 eleq2 2833 . . . . . 6 (𝑥 = ℕ → (1 ∈ 𝑥 ↔ 1 ∈ ℕ))
24 eleq2 2833 . . . . . . 7 (𝑥 = ℕ → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ ℕ))
2524raleqbi1dv 3294 . . . . . 6 (𝑥 = ℕ → (∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ))
2623, 25anbi12d 624 . . . . 5 (𝑥 = ℕ → ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ)))
2722, 26elab 3504 . . . 4 (ℕ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ))
2819, 21, 27mpbir2an 702 . . 3 ℕ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
29 intss1 4648 . . 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 3777 1 ℕ = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1650   = wceq 1652  wcel 2155  {cab 2751  wral 3055  wss 3732   cint 4633  (class class class)co 6842  1c1 10190   + caddc 10192  cn 11274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-1cn 10247  ax-addcl 10249
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-nn 11275
This theorem is referenced by:  dfnn3  11290
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