Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-dfom GIF version

Theorem bj-dfom 12965
 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 4474 . 2 ω = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
2 df-bj-ind 12959 . . . . 5 (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥))
32bicomi 131 . . . 4 ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) ↔ Ind 𝑥)
43abbii 2231 . . 3 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} = {𝑥 ∣ Ind 𝑥}
54inteqi 3743 . 2 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} = {𝑥 ∣ Ind 𝑥}
61, 5eqtri 2136 1 ω = {𝑥 ∣ Ind 𝑥}
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1314   ∈ wcel 1463  {cab 2101  ∀wral 2391  ∅c0 3331  ∩ cint 3739  suc csuc 4255  ωcom 4472  Ind wind 12958 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-int 3740  df-iom 4473  df-bj-ind 12959 This theorem is referenced by:  bj-omind  12966  bj-omssind  12967  bj-ssom  12968
 Copyright terms: Public domain W3C validator