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Theorem dfom3 9688
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 5306 . . . . 5 ∅ ∈ V
21elintab 4957 . . . 4 (∅ ∈ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → ∅ ∈ 𝑥))
3 simpl 482 . . . 4 ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → ∅ ∈ 𝑥)
42, 3mpgbir 1798 . . 3 ∅ ∈ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
5 suceq 6449 . . . . . . . . . 10 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
65eleq1d 2825 . . . . . . . . 9 (𝑦 = 𝑧 → (suc 𝑦𝑥 ↔ suc 𝑧𝑥))
76rspccv 3618 . . . . . . . 8 (∀𝑦𝑥 suc 𝑦𝑥 → (𝑧𝑥 → suc 𝑧𝑥))
87adantl 481 . . . . . . 7 ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → (𝑧𝑥 → suc 𝑧𝑥))
98a2i 14 . . . . . 6 (((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥) → ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → suc 𝑧𝑥))
109alimi 1810 . . . . 5 (∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥) → ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → suc 𝑧𝑥))
11 vex 3483 . . . . . 6 𝑧 ∈ V
1211elintab 4957 . . . . 5 (𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥))
1311sucex 7827 . . . . . 6 suc 𝑧 ∈ V
1413elintab 4957 . . . . 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 3064 . . 3 𝑧 ∈ ω (𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} → suc 𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)})
17 peano5 7916 . . 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 692 . 2 ω ⊆ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
19 peano1 7911 . . . 4 ∅ ∈ ω
20 peano2 7913 . . . . 5 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
2120rgen 3062 . . . 4 𝑦 ∈ ω suc 𝑦 ∈ ω
22 omex 9684 . . . . . 6 ω ∈ V
23 eleq2 2829 . . . . . . . 8 (𝑥 = ω → (∅ ∈ 𝑥 ↔ ∅ ∈ ω))
24 eleq2 2829 . . . . . . . . 9 (𝑥 = ω → (suc 𝑦𝑥 ↔ suc 𝑦 ∈ ω))
2524raleqbi1dv 3337 . . . . . . . 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 2829 . . . . . . 7 (𝑥 = ω → (𝑧𝑥𝑧 ∈ ω))
2826, 27imbi12d 344 . . . . . 6 (𝑥 = ω → (((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥) ↔ ((∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω) → 𝑧 ∈ ω)))
2922, 28spcv 3604 . . . . 5 (∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥) → ((∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω) → 𝑧 ∈ ω))
3012, 29sylbi 217 . . . 4 (𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} → ((∅ ∈ ω ∧ ∀𝑦 ∈ ω suc 𝑦 ∈ ω) → 𝑧 ∈ ω))
3119, 21, 30mp2ani 698 . . 3 (𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} → 𝑧 ∈ ω)
3231ssriv 3986 . 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 395  wal 1537   = wceq 1539  wcel 2107  {cab 2713  wral 3060  wss 3950  c0 4332   cint 4945  suc csuc 6385  ωcom 7888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-om 7889
This theorem is referenced by: (None)
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