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Theorem bj-dfom 13550
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 4552 . 2 ω = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
2 df-bj-ind 13544 . . . . 5 (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥))
32bicomi 131 . . . 4 ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) ↔ Ind 𝑥)
43abbii 2273 . . 3 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} = {𝑥 ∣ Ind 𝑥}
54inteqi 3812 . 2 {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)} = {𝑥 ∣ Ind 𝑥}
61, 5eqtri 2178 1 ω = {𝑥 ∣ Ind 𝑥}
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1335  wcel 2128  {cab 2143  wral 2435  c0 3394   cint 3808  suc csuc 4326  ωcom 4550  Ind wind 13543
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-int 3809  df-iom 4551  df-bj-ind 13544
This theorem is referenced by:  bj-omind  13551  bj-omssind  13552  bj-ssom  13553
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