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Theorem dfnn3 11641
Description: Alternate definition of the set of positive integers. Definition of positive integers in [Apostol] p. 22. (Contributed by NM, 3-Jul-2005.)
Assertion
Ref Expression
dfnn3 ℕ = {𝑥 ∣ (𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
Distinct variable group:   𝑥,𝑦

Proof of Theorem dfnn3
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2906 . . . 4 (𝑥 = 𝑧 → (1 ∈ 𝑥 ↔ 1 ∈ 𝑧))
2 eleq2 2906 . . . . 5 (𝑥 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ 𝑧))
32raleqbi1dv 3409 . . . 4 (𝑥 = 𝑧 → (∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧))
41, 3anbi12d 630 . . 3 (𝑥 = 𝑧 → ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)))
5 dfnn2 11640 . . . . 5 ℕ = {𝑧 ∣ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)}
65eqeq2i 2839 . . . 4 (𝑥 = ℕ ↔ 𝑥 = {𝑧 ∣ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)})
7 eleq2 2906 . . . . 5 (𝑥 = ℕ → (1 ∈ 𝑥 ↔ 1 ∈ ℕ))
8 eleq2 2906 . . . . . 6 (𝑥 = ℕ → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ ℕ))
98raleqbi1dv 3409 . . . . 5 (𝑥 = ℕ → (∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ))
107, 9anbi12d 630 . . . 4 (𝑥 = ℕ → ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ)))
116, 10sylbir 236 . . 3 (𝑥 = {𝑧 ∣ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)} → ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ)))
12 nnssre 11631 . . . . 5 ℕ ⊆ ℝ
135, 12eqsstrri 4006 . . . 4 {𝑧 ∣ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)} ⊆ ℝ
14 1nn 11638 . . . . 5 1 ∈ ℕ
15 peano2nn 11639 . . . . . 6 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
1615rgen 3153 . . . . 5 𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ
1714, 16pm3.2i 471 . . . 4 (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ)
1813, 17pm3.2i 471 . . 3 ( {𝑧 ∣ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)} ⊆ ℝ ∧ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ))
194, 11, 18intabs 5242 . 2 {𝑥 ∣ (𝑥 ⊆ ℝ ∧ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥))} = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
20 3anass 1089 . . . 4 ((𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (𝑥 ⊆ ℝ ∧ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)))
2120abbii 2891 . . 3 {𝑥 ∣ (𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} = {𝑥 ∣ (𝑥 ⊆ ℝ ∧ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥))}
2221inteqi 4878 . 2 {𝑥 ∣ (𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} = {𝑥 ∣ (𝑥 ⊆ ℝ ∧ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥))}
23 dfnn2 11640 . 2 ℕ = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
2419, 22, 233eqtr4ri 2860 1 ℕ = {𝑥 ∣ (𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  {cab 2804  wral 3143  wss 3940   cint 4874  (class class class)co 7148  cr 10525  1c1 10527   + caddc 10529  cn 11627
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 7451  ax-cnex 10582  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-i2m1 10594  ax-1ne0 10595  ax-rrecex 10598  ax-cnre 10599
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 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-nn 11628
This theorem is referenced by: (None)
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