Theorem elom 2375

Description: Membership in omega. The hypothesis can be eliminated if we assume the Axiom of Infinity; see elom3 3478.

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(s):   x,A

Proof of Theorem elom

Step	Hyp	Ref	Expression
1		elom.1	. 2 |- A e. V
2		ordeq 2206	. . 3 |- (y = A -> (Ord y <-> Ord A))
3		eleq1 1149	. . . . 5 |- (y = A -> (y e. x <-> A e. x))
4	3	imbi2d 464	. . . 4 |- (y = A -> ((Lim x -> y e. x) <-> (Lim x -> A e. x)))
5	4	bialdv 935	. . 3 |- (y = A -> (A.x(Lim x -> y e. x) <-> A.x(Lim x -> A e. x)))
6	2, 5	anbi12d 476	. 2 |- (y = A -> ((Ord y /\ A.x(Lim x -> y e. x)) <-> (Ord A /\ A.x(Lim x -> A e. x))))
7		df-om 2373	. 2 |- om = {y | (Ord y /\ A.x(Lim x -> y e. x))}
8	1, 6, 7	elab2 1419	1 |- (A e. om <-> (Ord A /\ A.x(Lim x -> A e. x)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672   e. wel 803   = wceq 1091   e. wcel 1092  Vcvv 1348  Ord word 2198  Lim wlim 2200  omcom 2372

This theorem is referenced by:  elomg 2376  omsson 2377  limomss 2378  ordom 2382

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-om 2373

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