Theorem bj-dfom 13302

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 4514	. 2 ⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}
2		df-bj-ind 13296	. . . . 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 2256	. . 3 ⊢ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} = {𝑥 ∣ Ind 𝑥}
5	4	inteqi 3783	. 2 ⊢ ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} = ∩ {𝑥 ∣ Ind 𝑥}
6	1, 5	eqtri 2161	1 ⊢ ω = ∩ {𝑥 ∣ Ind 𝑥}

Syntax hints:   ∧ wa 103   = wceq 1332   ∈ wcel 1481  {cab 2126  ∀wral 2417  ∅c0 3368  ∩ cint 3779  suc csuc 4295  ωcom 4512  Ind wind 13295

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-int 3780  df-iom 4513  df-bj-ind 13296

This theorem is referenced by:  bj-omind  13303  bj-omssind  13304  bj-ssom  13305