MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfnn3 Structured version   Visualization version   GIF version

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 2901 . . . 4 (𝑥 = 𝑧 → (1 ∈ 𝑥 ↔ 1 ∈ 𝑧))
2 eleq2 2901 . . . . 5 (𝑥 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ 𝑧))
32raleqbi1dv 3404 . . . 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 2834 . . . 4 (𝑥 = ℕ ↔ 𝑥 = {𝑧 ∣ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)})
7 eleq2 2901 . . . . 5 (𝑥 = ℕ → (1 ∈ 𝑥 ↔ 1 ∈ ℕ))
8 eleq2 2901 . . . . . 6 (𝑥 = ℕ → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ ℕ))
98raleqbi1dv 3404 . . . . 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 4001 . . . 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 3148 . . . . 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 5237 . 2 {𝑥 ∣ (𝑥 ⊆ ℝ ∧ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥))} = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
20 3anass 1087 . . . 4 ((𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (𝑥 ⊆ ℝ ∧ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)))
2120abbii 2886 . . 3 {𝑥 ∣ (𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} = {𝑥 ∣ (𝑥 ⊆ ℝ ∧ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥))}
2221inteqi 4873 . 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 2855 1 ℕ = {𝑥 ∣ (𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  {cab 2799  wral 3138  wss 3935   cint 4869  (class class class)co 7145  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 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  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 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7148  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-nn 11628
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator