Theorem bj-dfom 16351

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 4684	. 2 ⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}
2		df-bj-ind 16345	. . . . 5 ⊢ (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥))
3	2	bicomi 132	. . . 4 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) ↔ Ind 𝑥)
4	3	abbii 2345	. . 3 ⊢ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} = {𝑥 ∣ Ind 𝑥}
5	4	inteqi 3927	. 2 ⊢ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} = ∩ {𝑥 ∣ Ind 𝑥}
6	1, 5	eqtri 2250	1 ⊢ ω = ∩ {𝑥 ∣ Ind 𝑥}

Syntax hints:   ∧ wa 104   = wceq 1395   ∈ wcel 2200  {cab 2215  ∀wral 2508  ∅c0 3491  ∩ cint 3923  suc csuc 4456  ωcom 4682  Ind wind 16344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-int 3924  df-iom 4683  df-bj-ind 16345

This theorem is referenced by:  bj-omind  16352  bj-omssind  16353  bj-ssom  16354