Theorem elom 3134

Description: Membership in omega. The hypothesis can be eliminated if we assume the Axiom of Infinity; see elom3 4631.

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 2955	. . 3 |- (y = A -> (Ord y <-> Ord A))
3		eleq1 1534	. . . . 5 |- (y = A -> (y e. x <-> A e. x))
4	3	imbi2d 612	. . . 4 |- (y = A -> ((Lim x -> y e. x) <-> (Lim x -> A e. x)))
5	4	albidv 1278	. . 3 |- (y = A -> (A.x(Lim x -> y e. x) <-> A.x(Lim x -> A e. x)))
6	2, 5	anbi12d 628	. 2 |- (y = A -> ((Ord y /\ A.x(Lim x -> y e. x)) <-> (Ord A /\ A.x(Lim x -> A e. x))))
7		df-om 3132	. 2 |- om = {y | (Ord y /\ A.x(Lim x -> y e. x))}
8	1, 6, 7	elab2 1901	1 |- (A e. om <-> (Ord A /\ A.x(Lim x -> A e. x)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  Vcvv 1811  Ord word 2947  Lim wlim 2949  omcom 3131

This theorem is referenced by:  elomg 3135  omsson 3136  limomss 3137  ordom 3141

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-tr 2681  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-om 3132

Copyright terms: Public domain