Theorem elom 3221

Description: Membership in omega. The hypothesis can be eliminated if we assume the Axiom of Infinity; see elom3 4777.

Hypothesis

Ref	Expression
elom.1	|- A e. V

Assertion

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

Distinct variable group:   x,A

Proof of Theorem elom

Step	Hyp	Ref	Expression
1		elom.1	. 2 |- A e. V
2		ordeq 2982	. . 3 |- (y = A -> (Ord y <-> Ord A))
3		eleq1 1577	. . . . 5 |- (y = A -> (y e. x <-> A e. x))
4	3	imbi2d 615	. . . 4 |- (y = A -> ((Lim x -> y e. x) <-> (Lim x -> A e. x)))
5	4	albidv 1316	. . 3 |- (y = A -> (A.x(Lim x -> y e. x) <-> A.x(Lim x -> A e. x)))
6	2, 5	anbi12d 631	. 2 |- (y = A -> ((Ord y /\ A.x(Lim x -> y e. x)) <-> (Ord A /\ A.x(Lim x -> A e. x))))
7		df-om 3219	. 2 |- om = {y | (Ord y /\ A.x(Lim x -> y e. x))}
8	1, 6, 7	elab2 1947	1 |- (A e. om <-> (Ord A /\ A.x(Lim x -> A e. x)))

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221  A.wal 990   = wceq 992   e. wcel 994  Vcvv 1857  Ord word 2974  Lim wlim 2976  omcom 3218

This theorem is referenced by:  elomg 3222  omsson 3223  limomss 3224  ordom 3228

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 783  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-ral 1695  df-rex 1696  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-sn 2470  df-pr 2471  df-op 2474  df-uni 2570  df-br 2693  df-tr 2755  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-om 3219

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