Theorem limeq 6396

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 6391	. . 3 ⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
2		neeq1 3003	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ ∅ ↔ 𝐵 ≠ ∅))
3		id 22	. . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
4		unieq 4918	. . . 4 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
5	3, 4	eqeq12d 2753	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = ∪ 𝐴 ↔ 𝐵 = ∪ 𝐵))
6	1, 2, 5	3anbi123d 1438	. 2 ⊢ (𝐴 = 𝐵 → ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) ↔ (Ord 𝐵 ∧ 𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)))
7		df-lim 6389	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))
8		df-lim 6389	. 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 1540   ≠ wne 2940  ∅c0 4333  ∪ cuni 4907  Ord word 6383  Lim wlim 6385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-v 3482  df-ss 3968  df-uni 4908  df-tr 5260  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-lim 6389

This theorem is referenced by:  limuni2  6446  0ellim  6447  limuni3  7873  tfinds2  7885  dfom2  7889  limomss  7892  nnlim  7901  limom  7903  ssnlim  7907  onfununi  8381  tfr1a  8434  tz7.44lem1  8445  tz7.44-2  8447  tz7.44-3  8448  1ellim  8536  2ellim  8537  oeeulem  8639  limensuc  9194  elom3  9688  r1funlim  9806  rankxplim2  9920  rankxplim3  9921  rankxpsuc  9922  infxpenlem  10053  alephislim  10123  cflim2  10303  winalim  10735  rankcf  10817  gruina  10858  scutbdaybnd2lim  27862  rdgprc0  35794  dfrdg2  35796  dfrdg4  35952  limsucncmpi  36446  limsucncmp  36447  omlimcl2  43254  onexlimgt  43255  onov0suclim  43287  succlg  43341  dflim5  43342  nlim1NEW  43455  nlim2NEW  43456  nlim3  43457  nlim4  43458  dfsucon  43536