MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfnn2 Structured version   Visualization version   GIF version

Theorem dfnn2 11246
Description: Alternate definition of the set of positive integers. This was our original definition, before the current df-nn 11234 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 10248 . . . . 5 1 ∈ V
21elintab 4640 . . . 4 (1 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → 1 ∈ 𝑥))
3 simpl 474 . . . 4 ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → 1 ∈ 𝑥)
42, 3mpgbir 1875 . . 3 1 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
5 oveq1 6822 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑦 + 1) = (𝑧 + 1))
65eleq1d 2825 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑧 + 1) ∈ 𝑥))
76rspccv 3447 . . . . . . . 8 (∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥 → (𝑧𝑥 → (𝑧 + 1) ∈ 𝑥))
87adantl 473 . . . . . . 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 1888 . . . . 5 (∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → 𝑧𝑥) → ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → (𝑧 + 1) ∈ 𝑥))
11 vex 3344 . . . . . 6 𝑧 ∈ V
1211elintab 4640 . . . . 5 (𝑧 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → 𝑧𝑥))
13 ovex 6843 . . . . . 6 (𝑧 + 1) ∈ V
1413elintab 4640 . . . . 5 ((𝑧 + 1) ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑥((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → (𝑧 + 1) ∈ 𝑥))
1510, 12, 143imtr4i 281 . . . 4 (𝑧 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} → (𝑧 + 1) ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)})
1615rgen 3061 . . 3 𝑧 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} (𝑧 + 1) ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
17 peano5nni 11236 . . 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 710 . 2 ℕ ⊆ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
19 1nn 11244 . . . 4 1 ∈ ℕ
20 peano2nn 11245 . . . . 5 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
2120rgen 3061 . . . 4 𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ
22 nnex 11239 . . . . 5 ℕ ∈ V
23 eleq2 2829 . . . . . 6 (𝑥 = ℕ → (1 ∈ 𝑥 ↔ 1 ∈ ℕ))
24 eleq2 2829 . . . . . . 7 (𝑥 = ℕ → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ ℕ))
2524raleqbi1dv 3286 . . . . . 6 (𝑥 = ℕ → (∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ))
2623, 25anbi12d 749 . . . . 5 (𝑥 = ℕ → ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ)))
2722, 26elab 3491 . . . 4 (ℕ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ (1 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 + 1) ∈ ℕ))
2819, 21, 27mpbir2an 993 . . 3 ℕ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
29 intss1 4645 . . 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 3761 1 ℕ = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1630   = wceq 1632  wcel 2140  {cab 2747  wral 3051  wss 3716   cint 4628  (class class class)co 6815  1c1 10150   + caddc 10152  cn 11233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-i2m1 10217  ax-1ne0 10218  ax-rrecex 10221  ax-cnre 10222
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-om 7233  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-nn 11234
This theorem is referenced by:  dfnn3  11247
  Copyright terms: Public domain W3C validator