Theorem elom 3129

Description: Membership in omega. The hypothesis can be eliminated if we assume the Axiom of Infinity; see elom3 4611.

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 2950	. . 3 |- (y = A -> (Ord y <-> Ord A))
3		eleq1 1531	. . . . 5 |- (y = A -> (y e. x <-> A e. x))
4	3	imbi2d 611	. . . 4 |- (y = A -> ((Lim x -> y e. x) <-> (Lim x -> A e. x)))
5	4	albidv 1276	. . 3 |- (y = A -> (A.x(Lim x -> y e. x) <-> A.x(Lim x -> A e. x)))
6	2, 5	anbi12d 627	. 2 |- (y = A -> ((Ord y /\ A.x(Lim x -> y e. x)) <-> (Ord A /\ A.x(Lim x -> A e. x))))
7		df-om 3127	. 2 |- om = {y | (Ord y /\ A.x(Lim x -> y e. x))}
8	1, 6, 7	elab2 1897	1 |- (A e. om <-> (Ord A /\ A.x(Lim x -> A e. x)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  Vcvv 1807  Ord word 2942  Lim wlim 2944  omcom 3126

This theorem is referenced by:  elomg 3130  omsson 3131  limomss 3132  ordom 3136

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-tr 2676  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-om 3127

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