Theorem limeq 4059

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 4054	. . 3 ⊢ (A = B → (Ord A ↔ Ord B))
2		eleq2 2079	. . 3 ⊢ (A = B → (∅ ∈ A ↔ ∅ ∈ B))
3		id 19	. . . 4 ⊢ (A = B → A = B)
4		unieq 3559	. . . 4 ⊢ (A = B → ∪ A = ∪ B)
5	3, 4	eqeq12d 2032	. . 3 ⊢ (A = B → (A = ∪ A ↔ B = ∪ B))
6	1, 2, 5	3anbi123d 1190	. 2 ⊢ (A = B → ((Ord A ∧ ∅ ∈ A ∧ A = ∪ A) ↔ (Ord B ∧ ∅ ∈ B ∧ B = ∪ B)))
7		dflim2 4052	. 2 ⊢ (Lim A ↔ (Ord A ∧ ∅ ∈ A ∧ A = ∪ A))
8		dflim2 4052	. 2 ⊢ (Lim B ↔ (Ord B ∧ ∅ ∈ B ∧ B = ∪ B))
9	6, 7, 8	3bitr4g 212	1 ⊢ (A = B → (Lim A ↔ Lim B))

Syntax hints:   → wi 4   ↔ wb 98   ∧ w3a 871   = wceq 1226   ∈ wcel 1370  ∅c0 3197  ∪ cuni 3550  Ord word 4044  Lim wlim 4046

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-in 2897  df-ss 2904  df-uni 3551  df-tr 3825  df-iord 4048  df-ilim 4051

This theorem is referenced by:  limuni2  4079