Theorem limeq 6372

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 6367	. . 3 ⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
2		neeq1 3020	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ ∅ ↔ 𝐵 ≠ ∅))
3		id 23	. . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
4		unieq 4883	. . . 4 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
5	3, 4	eqeq12d 2779	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = ∪ 𝐴 ↔ 𝐵 = ∪ 𝐵))
6	1, 2, 5	3anbi123d 1464	. 2 ⊢ (𝐴 = 𝐵 → ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) ↔ (Ord 𝐵 ∧ 𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)))
7		df-lim 6365	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))
8		df-lim 6365	. 2 ⊢ (Lim 𝐵 ↔ (Ord 𝐵 ∧ 𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵))
9	6, 7, 8	3bitr4g 317	1 ⊢ (𝐴 = 𝐵 → (Lim 𝐴 ↔ Lim 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ≠ wne 2958  ∅c0 4286  ∪ cuni 4872  Ord word 6359  Lim wlim 6361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-v 3457  df-ss 3922  df-uni 4873  df-tr 5219  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-ord 6363  df-lim 6365

This theorem is referenced by:  limuni2  6424  limuni3  7844  tfinds2  7856  dfom2  7860  limomss  7863  nnlim  7872  limom  7874  ssnlim  7878  onfununi  8324  tfr1a  8377  tz7.44lem1  8388  tz7.44-2  8390  tz7.44-3  8391  1ellim  8479  2ellim  8480  oeeulem  8583  limensuc  9138  elom3  9613  r1funlim  9734  rankxplim2  9848  rankxplim3  9849  rankxpsuc  9850  infxpenlem  9993  alephislim  10063  cflim2  10242  winalim  10675  rankcf  10757  gruina  10798  cutbdaybnd2lim  27990  rdgprc0  36283  dfrdg2  36285  dfrdg4  36443  limsucncmpi  36956  limsucncmp  36957  omlimcl2  43969  onexlimgt  43970  onov0suclim  44001  succlg  44055  dflim5  44056  nlim1NEW  44168  nlim2NEW  44169  nlim3  44170  nlim4  44171  dfsucon  44249