Theorem bj-dfom 10728

Description: Alternate definition of om, as the intersection of all the inductive sets. Proposal: make this the definition. (Contributed by BJ, 30-Nov-2019.)

Assertion

Ref	Expression
bj-dfom	|- om = |^| { x | Ind x }

Proof of Theorem bj-dfom

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfom3 4333	. 2 |- om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) }
2		df-bj-ind 10722	. . . . 5 |- (Ind x <-> ( (/) e. x /\ A. y e. x suc y e. x ) )
3	2	bicomi 130	. . . 4 |- ( ( (/) e. x /\ A. y e. x suc y e. x ) <-> Ind x )
4	3	abbii 2194	. . 3 |- { x | ( (/) e. x /\ A. y e. x suc y e. x ) } = { x | Ind x }
5	4	inteqi 3640	. 2 |- |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) } = |^| { x | Ind x }
6	1, 5	eqtri 2101	1 |- om = |^| { x | Ind x }

Syntax hints:   /\ wa 102   = wceq 1284   e. wcel 1433   {cab 2067   A.wral 2348   (/)c0 3251   |^|cint 3636   suc csuc 4120   omcom 4331  Ind wind 10721

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-int 3637  df-iom 4332  df-bj-ind 10722

This theorem is referenced by:  bj-omind  10729  bj-omssind  10730  bj-ssom  10731