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Theorem dfnn2 10880
Description: Alternate definition of the set of positive integers. This was our original definition, before the current df-nn 10868 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 9891 . . . . 5 1 ∈ V
21elintab 4416 . . . 4 (1 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → 1 ∈ 𝑥))
3 simpl 471 . . . 4 ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → 1 ∈ 𝑥)
42, 3mpgbir 1716 . . 3 1 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
5 oveq1 6534 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑦 + 1) = (𝑧 + 1))
65eleq1d 2671 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑧 + 1) ∈ 𝑥))
76rspccv 3278 . . . . . . . 8 (∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥 → (𝑧𝑥 → (𝑧 + 1) ∈ 𝑥))
87adantl 480 . . . . . . 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 1729 . . . . 5 (∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → 𝑧𝑥) → ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → (𝑧 + 1) ∈ 𝑥))
11 vex 3175 . . . . . 6 𝑧 ∈ V
1211elintab 4416 . . . . 5 (𝑧 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → 𝑧𝑥))
13 ovex 6555 . . . . . 6 (𝑧 + 1) ∈ V
1413elintab 4416 . . . . 5 ((𝑧 + 1) ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → (𝑧 + 1) ∈ 𝑥))
1510, 12, 143imtr4i 279 . . . 4 (𝑧 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} → (𝑧 + 1) ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)})
1615rgen 2905 . . 3 𝑧 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} (𝑧 + 1) ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
17 peano5nni 10870 . . 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 703 . 2 ℕ ⊆ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
19 1nn 10878 . . . 4 1 ∈ ℕ
20 peano2nn 10879 . . . . 5 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
2120rgen 2905 . . . 4 𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ
22 nnex 10873 . . . . 5 ℕ ∈ V
23 eleq2 2676 . . . . . 6 (𝑥 = ℕ → (1 ∈ 𝑥 ↔ 1 ∈ ℕ))
24 eleq2 2676 . . . . . . 7 (𝑥 = ℕ → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ ℕ))
2524raleqbi1dv 3122 . . . . . 6 (𝑥 = ℕ → (∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ))
2623, 25anbi12d 742 . . . . 5 (𝑥 = ℕ → ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ)))
2722, 26elab 3318 . . . 4 (ℕ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ))
2819, 21, 27mpbir2an 956 . . 3 ℕ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
29 intss1 4421 . . 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 3583 1 ℕ = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1472   = wceq 1474  wcel 1976  {cab 2595  wral 2895  wss 3539   cint 4404  (class class class)co 6527  1c1 9793   + caddc 9795  cn 10867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-i2m1 9860  ax-1ne0 9861  ax-rrecex 9864  ax-cnre 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-om 6935  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-nn 10868
This theorem is referenced by:  dfnn3  10881
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