Theorem bj-dfom 13815

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 4569	. 2 |- om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) }
2		df-bj-ind 13809	. . . . 5 |- (Ind x <-> ( (/) e. x /\ A. y e. x suc y e. x ) )
3	2	bicomi 131	. . . 4 |- ( ( (/) e. x /\ A. y e. x suc y e. x ) <-> Ind x )
4	3	abbii 2282	. . 3 |- { x | ( (/) e. x /\ A. y e. x suc y e. x ) } = { x | Ind x }
5	4	inteqi 3828	. 2 |- |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) } = |^| { x | Ind x }
6	1, 5	eqtri 2186	1 |- om = |^| { x | Ind x }

Syntax hints:   /\ wa 103   = wceq 1343   e. wcel 2136   {cab 2151   A.wral 2444   (/)c0 3409   |^|cint 3824   suc csuc 4343   omcom 4567  Ind wind 13808

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-int 3825  df-iom 4568  df-bj-ind 13809

This theorem is referenced by:  bj-omind  13816  bj-omssind  13817  bj-ssom  13818