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Theorem bj-dfom 9321
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 4258 . 2 𝜔 = {x ∣ (∅ x y x suc y x)}
2 df-bj-ind 9316 . . . . 5 (Ind x ↔ (∅ x y x suc y x))
32bicomi 123 . . . 4 ((∅ x y x suc y x) ↔ Ind x)
43abbii 2150 . . 3 {x ∣ (∅ x y x suc y x)} = {x ∣ Ind x}
54inteqi 3610 . 2 {x ∣ (∅ x y x suc y x)} = {x ∣ Ind x}
61, 5eqtri 2057 1 𝜔 = {x ∣ Ind x}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242   wcel 1390  {cab 2023  wral 2300  c0 3218   cint 3606  suc csuc 4068  𝜔com 4256  Ind wind 9315
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-int 3607  df-iom 4257  df-bj-ind 9316
This theorem is referenced by:  bj-omind  9322  bj-omssind  9323  bj-ssom  9324
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