Theorem bj-dfom 16649

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 4696	. 2 |- om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) }
2		df-bj-ind 16643	. . . . 5 |- (Ind x <-> ( (/) e. x /\ A. y e. x suc y e. x ) )
3	2	bicomi 132	. . . 4 |- ( ( (/) e. x /\ A. y e. x suc y e. x ) <-> Ind x )
4	3	abbii 2347	. . 3 |- { x | ( (/) e. x /\ A. y e. x suc y e. x ) } = { x | Ind x }
5	4	inteqi 3937	. 2 |- |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) } = |^| { x | Ind x }
6	1, 5	eqtri 2252	1 |- om = |^| { x | Ind x }

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2202   {cab 2217   A.wral 2511   (/)c0 3496   |^|cint 3933   suc csuc 4468   omcom 4694  Ind wind 16642

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-int 3934  df-iom 4695  df-bj-ind 16643

This theorem is referenced by:  bj-omind  16650  bj-omssind  16651  bj-ssom  16652