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 → (∅ ∈ 𝐴 ↔ ∅ ∈ 𝐵))
3		id 19	. . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
4		unieq 3939	. . . 4 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
5	3, 4	eqeq12d 2253	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = ∪ 𝐴 ↔ 𝐵 = ∪ 𝐵))
6	1, 2, 5	3anbi123d 1353	. 2 ⊢ (𝐴 = 𝐵 → ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴) ↔ (Ord 𝐵 ∧ ∅ ∈ 𝐵 ∧ 𝐵 = ∪ 𝐵)))
7		dflim2 4510	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴))
8		dflim2 4510	. 2 ⊢ (Lim 𝐵 ↔ (Ord 𝐵 ∧ ∅ ∈ 𝐵 ∧ 𝐵 = ∪ 𝐵))
9	6, 7, 8	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (Lim 𝐴 ↔ Lim 𝐵))

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∅c0 3520  ∪ 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