Theorem limeq 4442

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 4437	. . 3 |- ( A = B -> ( Ord A <-> Ord B ) )
2		eleq2 2271	. . 3 |- ( A = B -> ( (/) e. A <-> (/) e. B ) )
3		id 19	. . . 4 |- ( A = B -> A = B )
4		unieq 3873	. . . 4 |- ( A = B -> U. A = U. B )
5	3, 4	eqeq12d 2222	. . 3 |- ( A = B -> ( A = U. A <-> B = U. B ) )
6	1, 2, 5	3anbi123d 1325	. 2 |- ( A = B -> ( ( Ord A /\ (/) e. A /\ A = U. A ) <-> ( Ord B /\ (/) e. B /\ B = U. B ) ) )
7		dflim2 4435	. 2 |- ( Lim A <-> ( Ord A /\ (/) e. A /\ A = U. A ) )
8		dflim2 4435	. 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 981   = wceq 1373   e. wcel 2178   (/)c0 3468   U.cuni 3864   Ord word 4427   Lim wlim 4429

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-in 3180  df-ss 3187  df-uni 3865  df-tr 4159  df-iord 4431  df-ilim 4434

This theorem is referenced by:  limuni2  4462