Theorem bj-dfom 10057

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 4315	. 2 |- om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) }
2		df-bj-ind 10051	. . . . 5 |- (Ind x <-> ( (/) e. x /\ A. y e. x suc y e. x ) )
3	2	bicomi 123	. . . 4 |- ( ( (/) e. x /\ A. y e. x suc y e. x ) <-> Ind x )
4	3	abbii 2153	. . 3 |- { x | ( (/) e. x /\ A. y e. x suc y e. x ) } = { x | Ind x }
5	4	inteqi 3619	. 2 |- |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) } = |^| { x | Ind x }
6	1, 5	eqtri 2060	1 |- om = |^| { x | Ind x }

Syntax hints:   /\ wa 97   = wceq 1243   e. wcel 1393   {cab 2026   A.wral 2306   (/)c0 3224   |^|cint 3615   suc csuc 4102   omcom 4313  Ind wind 10050

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-int 3616  df-iom 4314  df-bj-ind 10051

This theorem is referenced by:  bj-omind  10058  bj-omssind  10059  bj-ssom  10060  peano5setOLD  10065