Theorem limeq 4355

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 4350	. . 3 |- ( A = B -> ( Ord A <-> Ord B ) )
2		eleq2 2230	. . 3 |- ( A = B -> ( (/) e. A <-> (/) e. B ) )
3		id 19	. . . 4 |- ( A = B -> A = B )
4		unieq 3798	. . . 4 |- ( A = B -> U. A = U. B )
5	3, 4	eqeq12d 2180	. . 3 |- ( A = B -> ( A = U. A <-> B = U. B ) )
6	1, 2, 5	3anbi123d 1302	. 2 |- ( A = B -> ( ( Ord A /\ (/) e. A /\ A = U. A ) <-> ( Ord B /\ (/) e. B /\ B = U. B ) ) )
7		dflim2 4348	. 2 |- ( Lim A <-> ( Ord A /\ (/) e. A /\ A = U. A ) )
8		dflim2 4348	. 2 |- ( Lim B <-> ( Ord B /\ (/) e. B /\ B = U. B ) )
9	6, 7, 8	3bitr4g 222	1 |- ( A = B -> ( Lim A <-> Lim B ) )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   (/)c0 3409   U.cuni 3789   Ord word 4340   Lim wlim 4342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-in 3122  df-ss 3129  df-uni 3790  df-tr 4081  df-iord 4344  df-ilim 4347

This theorem is referenced by:  limuni2  4375