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Theorem dfom3 8488
Description: The class of natural numbers omega can be defined as the smallest "inductive set," 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 4750 . . . . 5 ∅ ∈ V
21elintab 4452 . . . 4 (∅ ∈ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → ∅ ∈ 𝑥))
3 simpl 473 . . . 4 ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → ∅ ∈ 𝑥)
42, 3mpgbir 1723 . . 3 ∅ ∈ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
5 suceq 5749 . . . . . . . . . 10 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
65eleq1d 2683 . . . . . . . . 9 (𝑦 = 𝑧 → (suc 𝑦𝑥 ↔ suc 𝑧𝑥))
76rspccv 3292 . . . . . . . 8 (∀𝑦𝑥 suc 𝑦𝑥 → (𝑧𝑥 → suc 𝑧𝑥))
87adantl 482 . . . . . . 7 ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → (𝑧𝑥 → suc 𝑧𝑥))
98a2i 14 . . . . . 6 (((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥) → ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → suc 𝑧𝑥))
109alimi 1736 . . . . 5 (∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥) → ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → suc 𝑧𝑥))
11 vex 3189 . . . . . 6 𝑧 ∈ V
1211elintab 4452 . . . . 5 (𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥))
1311sucex 6958 . . . . . 6 suc 𝑧 ∈ V
1413elintab 4452 . . . . 5 (suc 𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → suc 𝑧𝑥))
1510, 12, 143imtr4i 281 . . . 4 (𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} → suc 𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)})
1615rgenw 2919 . . 3 𝑧 ∈ ω (𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} → suc 𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)})
17 peano5 7036 . . 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 707 . 2 ω ⊆ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
19 peano1 7032 . . . 4 ∅ ∈ ω
20 peano2 7033 . . . . 5 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
2120rgen 2917 . . . 4 𝑦 ∈ ω suc 𝑦 ∈ ω
22 omex 8484 . . . . . 6 ω ∈ V
23 eleq2 2687 . . . . . . . 8 (𝑥 = ω → (∅ ∈ 𝑥 ↔ ∅ ∈ ω))
24 eleq2 2687 . . . . . . . . 9 (𝑥 = ω → (suc 𝑦𝑥 ↔ suc 𝑦 ∈ ω))
2524raleqbi1dv 3135 . . . . . . . 8 (𝑥 = ω → (∀𝑦𝑥 suc 𝑦𝑥 ↔ ∀𝑦 ∈ ω suc 𝑦 ∈ ω))
2623, 25anbi12d 746 . . . . . . 7 (𝑥 = ω → ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) ↔ (∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω)))
27 eleq2 2687 . . . . . . 7 (𝑥 = ω → (𝑧𝑥𝑧 ∈ ω))
2826, 27imbi12d 334 . . . . . 6 (𝑥 = ω → (((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥) ↔ ((∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω) → 𝑧 ∈ ω)))
2922, 28spcv 3285 . . . . 5 (∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥) → ((∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω) → 𝑧 ∈ ω))
3012, 29sylbi 207 . . . 4 (𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} → ((∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω) → 𝑧 ∈ ω))
3119, 21, 30mp2ani 713 . . 3 (𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} → 𝑧 ∈ ω)
3231ssriv 3587 . 2 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} ⊆ ω
3318, 32eqssi 3599 1 ω = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wss 3555  c0 3891   cint 4440  suc csuc 5684  ωcom 7012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-om 7013
This theorem is referenced by: (None)
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