Theorem limeq 6198

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 6193	. . 3 ⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
2		neeq1 3078	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ ∅ ↔ 𝐵 ≠ ∅))
3		id 22	. . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
4		unieq 4840	. . . 4 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
5	3, 4	eqeq12d 2837	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = ∪ 𝐴 ↔ 𝐵 = ∪ 𝐵))
6	1, 2, 5	3anbi123d 1432	. 2 ⊢ (𝐴 = 𝐵 → ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) ↔ (Ord 𝐵 ∧ 𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)))
7		df-lim 6191	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))
8		df-lim 6191	. 2 ⊢ (Lim 𝐵 ↔ (Ord 𝐵 ∧ 𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵))
9	6, 7, 8	3bitr4g 316	1 ⊢ (𝐴 = 𝐵 → (Lim 𝐴 ↔ Lim 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1083   = wceq 1533   ≠ wne 3016  ∅c0 4291  ∪ cuni 4832  Ord word 6185  Lim wlim 6187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-ne 3017  df-ral 3143  df-rex 3144  df-in 3943  df-ss 3952  df-uni 4833  df-tr 5166  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-ord 6189  df-lim 6191

This theorem is referenced by:  limuni2  6247  0ellim  6248  limuni3  7561  tfinds2  7572  dfom2  7576  limomss  7579  nnlim  7587  limom  7589  ssnlim  7593  onfununi  7972  tfr1a  8024  tz7.44lem1  8035  tz7.44-2  8037  tz7.44-3  8038  oeeulem  8221  limensuc  8688  elom3  9105  r1funlim  9189  rankxplim2  9303  rankxplim3  9304  rankxpsuc  9305  infxpenlem  9433  alephislim  9503  cflim2  9679  winalim  10111  rankcf  10193  gruina  10234  rdgprc0  33033  dfrdg2  33035  dfrdg4  33407  limsucncmpi  33788  limsucncmp  33789  dfsucon  39882