Theorem limeq 4498

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 4493	. . 3 |- ( A = B -> ( Ord A <-> Ord B ) )
2		eleq2 2296	. . 3 |- ( A = B -> ( (/) e. A <-> (/) e. B ) )
3		id 19	. . . 4 |- ( A = B -> A = B )
4		unieq 3923	. . . 4 |- ( A = B -> U. A = U. B )
5	3, 4	eqeq12d 2247	. . 3 |- ( A = B -> ( A = U. A <-> B = U. B ) )
6	1, 2, 5	3anbi123d 1349	. 2 |- ( A = B -> ( ( Ord A /\ (/) e. A /\ A = U. A ) <-> ( Ord B /\ (/) e. B /\ B = U. B ) ) )
7		dflim2 4491	. 2 |- ( Lim A <-> ( Ord A /\ (/) e. A /\ A = U. A ) )
8		dflim2 4491	. 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 1005   = wceq 1398   e. wcel 2203   (/)c0 3508   U.cuni 3914   Ord word 4483   Lim wlim 4485

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-in 3217  df-ss 3224  df-uni 3915  df-tr 4209  df-iord 4487  df-ilim 4490

This theorem is referenced by:  limuni2  4518