Theorem elom 7068

Description: Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 8545. (Contributed by NM, 15-May-1994.)

Assertion

Ref	Expression
elom	|- ( A e. om <-> ( A e. On /\ A. x ( Lim x -> A e. x ) ) )

Distinct variable group:   x, A

Proof of Theorem elom

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . 4 |- ( y = A -> ( y e. x <-> A e. x ) )
2	1	imbi2d 330	. . 3 |- ( y = A -> ( ( Lim x -> y e. x ) <-> ( Lim x -> A e. x ) ) )
3	2	albidv 1849	. 2 |- ( y = A -> ( A. x ( Lim x -> y e. x ) <-> A. x ( Lim x -> A e. x ) ) )
4		df-om 7066	. 2 |- om = { y e. On | A. x ( Lim x -> y e. x ) }
5	3, 4	elrab2 3366	1 |- ( A e. om <-> ( A e. On /\ A. x ( Lim x -> A e. x ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   Oncon0 5723   Lim wlim 5724   omcom 7065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-om 7066

This theorem is referenced by:  limomss  7070  ordom  7074  nnlim  7078  limom  7080  elom3  8545  dfom5b  32019