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Theorem dfom3 9685
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 5313 . . . . 5 ∅ ∈ V
21elintab 4963 . . . 4 (∅ ∈ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → ∅ ∈ 𝑥))
3 simpl 482 . . . 4 ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → ∅ ∈ 𝑥)
42, 3mpgbir 1796 . . 3 ∅ ∈ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
5 suceq 6452 . . . . . . . . . 10 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
65eleq1d 2824 . . . . . . . . 9 (𝑦 = 𝑧 → (suc 𝑦𝑥 ↔ suc 𝑧𝑥))
76rspccv 3619 . . . . . . . 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 1808 . . . . 5 (∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥) → ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → suc 𝑧𝑥))
11 vex 3482 . . . . . 6 𝑧 ∈ V
1211elintab 4963 . . . . 5 (𝑧 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} ↔ ∀𝑥((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) → 𝑧𝑥))
1311sucex 7826 . . . . . 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 3063 . . 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 3061 . . . 4 𝑦 ∈ ω suc 𝑦 ∈ ω
22 omex 9681 . . . . . 6 ω ∈ V
23 eleq2 2828 . . . . . . . 8 (𝑥 = ω → (∅ ∈ 𝑥 ↔ ∅ ∈ ω))
24 eleq2 2828 . . . . . . . . 9 (𝑥 = ω → (suc 𝑦𝑥 ↔ suc 𝑦 ∈ ω))
2524raleqbi1dv 3336 . . . . . . . 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 2828 . . . . . . 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 3605 . . . . 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 3999 . 2 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} ⊆ ω
3318, 32eqssi 4012 1 ω = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wss 3963  c0 4339   cint 4951  suc csuc 6388  ωcom 7887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-om 7888
This theorem is referenced by: (None)
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