Theorem bj-dfom 15906

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 4641	. 2 |- om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) }
2		df-bj-ind 15900	. . . . 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 2321	. . 3 |- { x | ( (/) e. x /\ A. y e. x suc y e. x ) } = { x | Ind x }
5	4	inteqi 3889	. 2 |- |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) } = |^| { x | Ind x }
6	1, 5	eqtri 2226	1 |- om = |^| { x | Ind x }

Syntax hints:   /\ wa 104   = wceq 1373   e. wcel 2176   {cab 2191   A.wral 2484   (/)c0 3460   |^|cint 3885   suc csuc 4413   omcom 4639  Ind wind 15899

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-int 3886  df-iom 4640  df-bj-ind 15900

This theorem is referenced by:  bj-omind  15907  bj-omssind  15908  bj-ssom  15909