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Theorem bj-dfom 7155
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 4242 . 2 𝜔 = {x ∣ (∅ x y x suc y x)}
2 df-bj-ind 7150 . . . . 5 (Ind x ↔ (∅ x y x suc y x))
32bicomi 123 . . . 4 ((∅ x y x suc y x) ↔ Ind x)
43abbii 2135 . . 3 {x ∣ (∅ x y x suc y x)} = {x ∣ Ind x}
54inteqi 3593 . 2 {x ∣ (∅ x y x suc y x)} = {x ∣ Ind x}
61, 5eqtri 2042 1 𝜔 = {x ∣ Ind x}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1228   wcel 1374  {cab 2008  wral 2284  c0 3201   cint 3589  suc csuc 4051  𝜔com 4240  Ind wind 7149
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-int 3590  df-iom 4241  df-bj-ind 7150
This theorem is referenced by:  bj-omind  7156  bj-omssind  7157  bj-ssom  7158
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