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

Theorem dfom3 9107
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. (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 5197 . . . . 5 ∅ ∈ V
21elintab 4873 . . . 4 (∅ ∈ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → ∅ ∈ 𝑥))
3 simpl 486 . . . 4 ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → ∅ ∈ 𝑥)
42, 3mpgbir 1801 . . 3 ∅ ∈ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
5 suceq 6243 . . . . . . . . . 10 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
65eleq1d 2900 . . . . . . . . 9 (𝑦 = 𝑧 → (suc 𝑦𝑥 ↔ suc 𝑧𝑥))
76rspccv 3606 . . . . . . . 8 (∀𝑦𝑥 suc 𝑦𝑥 → (𝑧𝑥 → suc 𝑧𝑥))
87adantl 485 . . . . . . 7 ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → (𝑧𝑥 → suc 𝑧𝑥))
98a2i 14 . . . . . 6 (((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥) → ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → suc 𝑧𝑥))
109alimi 1813 . . . . 5 (∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥) → ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → suc 𝑧𝑥))
11 vex 3483 . . . . . 6 𝑧 ∈ V
1211elintab 4873 . . . . 5 (𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥))
1311sucex 7520 . . . . . 6 suc 𝑧 ∈ V
1413elintab 4873 . . . . 5 (suc 𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → suc 𝑧𝑥))
1510, 12, 143imtr4i 295 . . . 4 (𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} → suc 𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)})
1615rgenw 3145 . . 3 𝑧 ∈ ω (𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} → suc 𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)})
17 peano5 7599 . . 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 7595 . . . 4 ∅ ∈ ω
20 peano2 7596 . . . . 5 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
2120rgen 3143 . . . 4 𝑦 ∈ ω suc 𝑦 ∈ ω
22 omex 9103 . . . . . 6 ω ∈ V
23 eleq2 2904 . . . . . . . 8 (𝑥 = ω → (∅ ∈ 𝑥 ↔ ∅ ∈ ω))
24 eleq2 2904 . . . . . . . . 9 (𝑥 = ω → (suc 𝑦𝑥 ↔ suc 𝑦 ∈ ω))
2524raleqbi1dv 3394 . . . . . . . 8 (𝑥 = ω → (∀𝑦𝑥 suc 𝑦𝑥 ↔ ∀𝑦 ∈ ω suc 𝑦 ∈ ω))
2623, 25anbi12d 633 . . . . . . 7 (𝑥 = ω → ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) ↔ (∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω)))
27 eleq2 2904 . . . . . . 7 (𝑥 = ω → (𝑧𝑥𝑧 ∈ ω))
2826, 27imbi12d 348 . . . . . 6 (𝑥 = ω → (((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥) ↔ ((∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω) → 𝑧 ∈ ω)))
2922, 28spcv 3592 . . . . 5 (∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥) → ((∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω) → 𝑧 ∈ ω))
3012, 29sylbi 220 . . . 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 3957 . 2 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} ⊆ ω
3318, 32eqssi 3969 1 ω = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536   = wceq 1538  wcel 2115  {cab 2802  wral 3133  wss 3919  c0 4276   cint 4862  suc csuc 6180  ωcom 7574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455  ax-inf2 9101
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-br 5053  df-opab 5115  df-tr 5159  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-om 7575
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator