Theorem bj-dfom 13131

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.

Step	Hyp	Ref	Expression
1		dfom3 4506	. 2 ⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}
2		df-bj-ind 13125	. . . . 5 ⊢ (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥))
3	2	bicomi 131	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) ↔ Ind 𝑥)
4	3	abbii 2255	. . 3 ⊢ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} = {𝑥 ∣ Ind 𝑥}
5	4	inteqi 3775	. 2 ⊢ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} = ∩ {𝑥 ∣ Ind 𝑥}
6	1, 5	eqtri 2160	1 ⊢ ω = ∩ {𝑥 ∣ Ind 𝑥}

Syntax hints:   ∧ wa 103   = wceq 1331   ∈ wcel 1480  {cab 2125  ∀wral 2416  ∅c0 3363  ∩ cint 3771  suc csuc 4287  ωcom 4504  Ind wind 13124

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-int 3772  df-iom 4505  df-bj-ind 13125

This theorem is referenced by:  bj-omind  13132  bj-omssind  13133  bj-ssom  13134