Theorem limeq 6335

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 6330	. . 3 ⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
2		neeq1 2994	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ ∅ ↔ 𝐵 ≠ ∅))
3		id 22	. . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
4		unieq 4861	. . . 4 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
5	3, 4	eqeq12d 2752	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = ∪ 𝐴 ↔ 𝐵 = ∪ 𝐵))
6	1, 2, 5	3anbi123d 1439	. 2 ⊢ (𝐴 = 𝐵 → ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) ↔ (Ord 𝐵 ∧ 𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)))
7		df-lim 6328	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))
8		df-lim 6328	. 2 ⊢ (Lim 𝐵 ↔ (Ord 𝐵 ∧ 𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵))
9	6, 7, 8	3bitr4g 314	1 ⊢ (𝐴 = 𝐵 → (Lim 𝐴 ↔ Lim 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1542   ≠ wne 2932  ∅c0 4273  ∪ cuni 4850  Ord word 6322  Lim wlim 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-v 3431  df-ss 3906  df-uni 4851  df-tr 5193  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6326  df-lim 6328

This theorem is referenced by:  limuni2  6386  limuni3  7803  tfinds2  7815  dfom2  7819  limomss  7822  nnlim  7831  limom  7833  ssnlim  7837  onfununi  8281  tfr1a  8333  tz7.44lem1  8344  tz7.44-2  8346  tz7.44-3  8347  1ellim  8433  2ellim  8434  oeeulem  8537  limensuc  9092  elom3  9569  r1funlim  9690  rankxplim2  9804  rankxplim3  9805  rankxpsuc  9806  infxpenlem  9935  alephislim  10005  cflim2  10185  winalim  10618  rankcf  10700  gruina  10741  cutbdaybnd2lim  27789  rdgprc0  35973  dfrdg2  35975  dfrdg4  36133  limsucncmpi  36627  limsucncmp  36628  omlimcl2  43670  onexlimgt  43671  onov0suclim  43702  succlg  43756  dflim5  43757  nlim1NEW  43869  nlim2NEW  43870  nlim3  43871  nlim4  43872  dfsucon  43950