Theorem limeq 4517

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 4512	. . 3 |- ( A = B -> ( Ord A <-> Ord B ) )
2		eleq2 2302	. . 3 |- ( A = B -> ( (/) e. A <-> (/) e. B ) )
3		id 19	. . . 4 |- ( A = B -> A = B )
4		unieq 3939	. . . 4 |- ( A = B -> U. A = U. B )
5	3, 4	eqeq12d 2253	. . 3 |- ( A = B -> ( A = U. A <-> B = U. B ) )
6	1, 2, 5	3anbi123d 1353	. 2 |- ( A = B -> ( ( Ord A /\ (/) e. A /\ A = U. A ) <-> ( Ord B /\ (/) e. B /\ B = U. B ) ) )
7		dflim2 4510	. 2 |- ( Lim A <-> ( Ord A /\ (/) e. A /\ A = U. A ) )
8		dflim2 4510	. 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 1009   = wceq 1402   e. wcel 2209   (/)c0 3520   U.cuni 3930   Ord word 4502   Lim wlim 4504

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-in 3226  df-ss 3233  df-uni 3931  df-tr 4225  df-iord 4506  df-ilim 4509

This theorem is referenced by:  limuni2  4537