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Theorem bj-dfom 11474
Description: Alternate definition of ω, as the intersection of all the inductive sets. Proposal: make this the definition. (Contributed by BJ, 30-Nov-2019.)
Assertion
Ref Expression
bj-dfom ω = {𝑥 ∣ Ind 𝑥}

Proof of Theorem bj-dfom
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfom3 4397 . 2 ω = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
2 df-bj-ind 11468 . . . . 5 (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥))
32bicomi 130 . . . 4 ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) ↔ Ind 𝑥)
43abbii 2203 . . 3 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} = {𝑥 ∣ Ind 𝑥}
54inteqi 3687 . 2 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} = {𝑥 ∣ Ind 𝑥}
61, 5eqtri 2108 1 ω = {𝑥 ∣ Ind 𝑥}
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1289  wcel 1438  {cab 2074  wral 2359  c0 3284   cint 3683  suc csuc 4183  ωcom 4395  Ind wind 11467
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-int 3684  df-iom 4396  df-bj-ind 11468
This theorem is referenced by:  bj-omind  11475  bj-omssind  11476  bj-ssom  11477
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