Theorem limeq 4379

Description: Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
limeq	|- ( A = B -> ( Lim A <-> Lim B ) )

Proof of Theorem limeq

Step	Hyp	Ref	Expression
1		ordeq 4374	. . 3 |- ( A = B -> ( Ord A <-> Ord B ) )
2		eleq2 2241	. . 3 |- ( A = B -> ( (/) e. A <-> (/) e. B ) )
3		id 19	. . . 4 |- ( A = B -> A = B )
4		unieq 3820	. . . 4 |- ( A = B -> U. A = U. B )
5	3, 4	eqeq12d 2192	. . 3 |- ( A = B -> ( A = U. A <-> B = U. B ) )
6	1, 2, 5	3anbi123d 1312	. 2 |- ( A = B -> ( ( Ord A /\ (/) e. A /\ A = U. A ) <-> ( Ord B /\ (/) e. B /\ B = U. B ) ) )
7		dflim2 4372	. 2 |- ( Lim A <-> ( Ord A /\ (/) e. A /\ A = U. A ) )
8		dflim2 4372	. 2 |- ( Lim B <-> ( Ord B /\ (/) e. B /\ B = U. B ) )
9	6, 7, 8	3bitr4g 223	1 |- ( A = B -> ( Lim A <-> Lim B ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   (/)c0 3424   U.cuni 3811   Ord word 4364   Lim wlim 4366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-in 3137  df-ss 3144  df-uni 3812  df-tr 4104  df-iord 4368  df-ilim 4371

This theorem is referenced by:  limuni2  4399