MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfom3 Structured version   Visualization version   GIF version

Theorem dfom3 9642
Description: The class of natural numbers ω can be defined as the intersection of all inductive sets (which is the smallest inductive set, since inductive sets are closed under intersection), which is valid provided we assume the Axiom of Infinity. Definition 6.3 of [Eisenberg] p. 82. Definition 1.20 of [Schloeder] p. 3. (Contributed by NM, 6-Aug-1994.)
Assertion
Ref Expression
dfom3 ω = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
Distinct variable group:   𝑥,𝑦

Proof of Theorem dfom3
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 0ex 5308 . . . . 5 ∅ ∈ V
21elintab 4963 . . . 4 (∅ ∈ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → ∅ ∈ 𝑥))
3 simpl 484 . . . 4 ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → ∅ ∈ 𝑥)
42, 3mpgbir 1802 . . 3 ∅ ∈ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
5 suceq 6431 . . . . . . . . . 10 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
65eleq1d 2819 . . . . . . . . 9 (𝑦 = 𝑧 → (suc 𝑦𝑥 ↔ suc 𝑧𝑥))
76rspccv 3610 . . . . . . . 8 (∀𝑦𝑥 suc 𝑦𝑥 → (𝑧𝑥 → suc 𝑧𝑥))
87adantl 483 . . . . . . 7 ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → (𝑧𝑥 → suc 𝑧𝑥))
98a2i 14 . . . . . 6 (((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥) → ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → suc 𝑧𝑥))
109alimi 1814 . . . . 5 (∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥) → ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → suc 𝑧𝑥))
11 vex 3479 . . . . . 6 𝑧 ∈ V
1211elintab 4963 . . . . 5 (𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥))
1311sucex 7794 . . . . . 6 suc 𝑧 ∈ V
1413elintab 4963 . . . . 5 (suc 𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → suc 𝑧𝑥))
1510, 12, 143imtr4i 292 . . . 4 (𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} → suc 𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)})
1615rgenw 3066 . . 3 𝑧 ∈ ω (𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} → suc 𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)})
17 peano5 7884 . . 3 ((∅ ∈ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} ∧ ∀𝑧 ∈ ω (𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} → suc 𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)})) → ω ⊆ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)})
184, 16, 17mp2an 691 . 2 ω ⊆ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
19 peano1 7879 . . . 4 ∅ ∈ ω
20 peano2 7881 . . . . 5 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
2120rgen 3064 . . . 4 𝑦 ∈ ω suc 𝑦 ∈ ω
22 omex 9638 . . . . . 6 ω ∈ V
23 eleq2 2823 . . . . . . . 8 (𝑥 = ω → (∅ ∈ 𝑥 ↔ ∅ ∈ ω))
24 eleq2 2823 . . . . . . . . 9 (𝑥 = ω → (suc 𝑦𝑥 ↔ suc 𝑦 ∈ ω))
2524raleqbi1dv 3334 . . . . . . . 8 (𝑥 = ω → (∀𝑦𝑥 suc 𝑦𝑥 ↔ ∀𝑦 ∈ ω suc 𝑦 ∈ ω))
2623, 25anbi12d 632 . . . . . . 7 (𝑥 = ω → ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) ↔ (∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω)))
27 eleq2 2823 . . . . . . 7 (𝑥 = ω → (𝑧𝑥𝑧 ∈ ω))
2826, 27imbi12d 345 . . . . . 6 (𝑥 = ω → (((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥) ↔ ((∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω) → 𝑧 ∈ ω)))
2922, 28spcv 3596 . . . . 5 (∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥) → ((∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω) → 𝑧 ∈ ω))
3012, 29sylbi 216 . . . 4 (𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} → ((∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω) → 𝑧 ∈ ω))
3119, 21, 30mp2ani 697 . . 3 (𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} → 𝑧 ∈ ω)
3231ssriv 3987 . 2 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} ⊆ ω
3318, 32eqssi 3999 1 ω = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540   = wceq 1542  wcel 2107  {cab 2710  wral 3062  wss 3949  c0 4323   cint 4951  suc csuc 6367  ωcom 7855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725  ax-inf2 9636
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-om 7856
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator