Theorem limeq 4362

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 4357	. . 3 ⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
2		eleq2 2234	. . 3 ⊢ (𝐴 = 𝐵 → (∅ ∈ 𝐴 ↔ ∅ ∈ 𝐵))
3		id 19	. . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
4		unieq 3805	. . . 4 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
5	3, 4	eqeq12d 2185	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = ∪ 𝐴 ↔ 𝐵 = ∪ 𝐵))
6	1, 2, 5	3anbi123d 1307	. 2 ⊢ (𝐴 = 𝐵 → ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴) ↔ (Ord 𝐵 ∧ ∅ ∈ 𝐵 ∧ 𝐵 = ∪ 𝐵)))
7		dflim2 4355	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴))
8		dflim2 4355	. 2 ⊢ (Lim 𝐵 ↔ (Ord 𝐵 ∧ ∅ ∈ 𝐵 ∧ 𝐵 = ∪ 𝐵))
9	6, 7, 8	3bitr4g 222	1 ⊢ (𝐴 = 𝐵 → (Lim 𝐴 ↔ Lim 𝐵))

Syntax hints:   → wi 4   ↔ wb 104   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  ∅c0 3414  ∪ cuni 3796  Ord word 4347  Lim wlim 4349

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088  df-iord 4351  df-ilim 4354

This theorem is referenced by:  limuni2  4382