Theorem limeq 4474

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 4469	. . 3 ⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
2		eleq2 2295	. . 3 ⊢ (𝐴 = 𝐵 → (∅ ∈ 𝐴 ↔ ∅ ∈ 𝐵))
3		id 19	. . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
4		unieq 3902	. . . 4 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
5	3, 4	eqeq12d 2246	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = ∪ 𝐴 ↔ 𝐵 = ∪ 𝐵))
6	1, 2, 5	3anbi123d 1348	. 2 ⊢ (𝐴 = 𝐵 → ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴) ↔ (Ord 𝐵 ∧ ∅ ∈ 𝐵 ∧ 𝐵 = ∪ 𝐵)))
7		dflim2 4467	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴))
8		dflim2 4467	. 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 1004   = wceq 1397   ∈ wcel 2202  ∅c0 3494  ∪ cuni 3893  Ord word 4459  Lim wlim 4461

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-in 3206  df-ss 3213  df-uni 3894  df-tr 4188  df-iord 4463  df-ilim 4466

This theorem is referenced by:  limuni2  4494