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Theorem dfnn3 12262
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 2817 . . . 4 (𝑥 = 𝑧 → (1 ∈ 𝑥 ↔ 1 ∈ 𝑧))
2 eleq2 2817 . . . . 5 (𝑥 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ 𝑧))
32raleqbi1dv 3329 . . . 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 12261 . . . . 5 ℕ = {𝑧 ∣ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)}
65eqeq2i 2740 . . . 4 (𝑥 = ℕ ↔ 𝑥 = {𝑧 ∣ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)})
7 eleq2 2817 . . . . 5 (𝑥 = ℕ → (1 ∈ 𝑥 ↔ 1 ∈ ℕ))
8 eleq2 2817 . . . . . 6 (𝑥 = ℕ → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ ℕ))
98raleqbi1dv 3329 . . . . 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 234 . . 3 (𝑥 = {𝑧 ∣ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)} → ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ)))
12 nnssre 12252 . . . . 5 ℕ ⊆ ℝ
135, 12eqsstrri 4015 . . . 4 {𝑧 ∣ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)} ⊆ ℝ
14 1nn 12259 . . . . 5 1 ∈ ℕ
15 peano2nn 12260 . . . . . 6 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
1615rgen 3059 . . . . 5 𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ
1714, 16pm3.2i 469 . . . 4 (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ)
1813, 17pm3.2i 469 . . 3 ( {𝑧 ∣ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)} ⊆ ℝ ∧ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ))
194, 11, 18intabs 5346 . 2 {𝑥 ∣ (𝑥 ⊆ ℝ ∧ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥))} = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
20 3anass 1092 . . . 4 ((𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (𝑥 ⊆ ℝ ∧ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)))
2120abbii 2797 . . 3 {𝑥 ∣ (𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} = {𝑥 ∣ (𝑥 ⊆ ℝ ∧ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥))}
2221inteqi 4955 . 2 {𝑥 ∣ (𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} = {𝑥 ∣ (𝑥 ⊆ ℝ ∧ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥))}
23 dfnn2 12261 . 2 ℕ = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
2419, 22, 233eqtr4ri 2766 1 ℕ = {𝑥 ∣ (𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  {cab 2704  wral 3057  wss 3947   cint 4951  (class class class)co 7424  cr 11143  1c1 11145   + caddc 11147  cn 12248
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744  ax-cnex 11200  ax-1cn 11202  ax-icn 11203  ax-addcl 11204  ax-addrcl 11205  ax-mulcl 11206  ax-mulrcl 11207  ax-i2m1 11212  ax-1ne0 11213  ax-rrecex 11216  ax-cnre 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-om 7875  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-nn 12249
This theorem is referenced by: (None)
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