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Theorem dfnn2 11645
Description: Alternate definition of the set of positive integers. This was our original definition, before the current df-nn 11633 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 10631 . . . . 5 1 ∈ V
21elintab 4885 . . . 4 (1 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → 1 ∈ 𝑥))
3 simpl 483 . . . 4 ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → 1 ∈ 𝑥)
42, 3mpgbir 1793 . . 3 1 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
5 oveq1 7157 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑦 + 1) = (𝑧 + 1))
65eleq1d 2902 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑧 + 1) ∈ 𝑥))
76rspccv 3624 . . . . . . . 8 (∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥 → (𝑧𝑥 → (𝑧 + 1) ∈ 𝑥))
87adantl 482 . . . . . . 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 1805 . . . . 5 (∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → 𝑧𝑥) → ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → (𝑧 + 1) ∈ 𝑥))
11 vex 3503 . . . . . 6 𝑧 ∈ V
1211elintab 4885 . . . . 5 (𝑧 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → 𝑧𝑥))
13 ovex 7183 . . . . . 6 (𝑧 + 1) ∈ V
1413elintab 4885 . . . . 5 ((𝑧 + 1) ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → (𝑧 + 1) ∈ 𝑥))
1510, 12, 143imtr4i 293 . . . 4 (𝑧 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} → (𝑧 + 1) ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)})
1615rgen 3153 . . 3 𝑧 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} (𝑧 + 1) ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
17 peano5nni 11635 . . 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 688 . 2 ℕ ⊆ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
19 1nn 11643 . . . 4 1 ∈ ℕ
20 peano2nn 11644 . . . . 5 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
2120rgen 3153 . . . 4 𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ
22 nnex 11638 . . . . 5 ℕ ∈ V
23 eleq2 2906 . . . . . 6 (𝑥 = ℕ → (1 ∈ 𝑥 ↔ 1 ∈ ℕ))
24 eleq2 2906 . . . . . . 7 (𝑥 = ℕ → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ ℕ))
2524raleqbi1dv 3409 . . . . . 6 (𝑥 = ℕ → (∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ))
2623, 25anbi12d 630 . . . . 5 (𝑥 = ℕ → ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ)))
2722, 26elab 3671 . . . 4 (ℕ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ))
2819, 21, 27mpbir2an 707 . . 3 ℕ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
29 intss1 4889 . . 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 3987 1 ℕ = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1528   = wceq 1530  wcel 2107  {cab 2804  wral 3143  wss 3940   cint 4874  (class class class)co 7150  1c1 10532   + caddc 10534  cn 11632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-1cn 10589  ax-addcl 10591
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-om 7574  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-nn 11633
This theorem is referenced by:  dfnn3  11646
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