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Mirrors > Home > MPE Home > Th. List > limeq | Structured version Visualization version GIF version |
Description: Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
limeq | ⊢ (𝐴 = 𝐵 → (Lim 𝐴 ↔ Lim 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordeq 6392 | . . 3 ⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵)) | |
2 | neeq1 3000 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ ∅ ↔ 𝐵 ≠ ∅)) | |
3 | id 22 | . . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | |
4 | unieq 4922 | . . . 4 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵) | |
5 | 3, 4 | eqeq12d 2750 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = ∪ 𝐴 ↔ 𝐵 = ∪ 𝐵)) |
6 | 1, 2, 5 | 3anbi123d 1435 | . 2 ⊢ (𝐴 = 𝐵 → ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) ↔ (Ord 𝐵 ∧ 𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵))) |
7 | df-lim 6390 | . 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | |
8 | df-lim 6390 | . 2 ⊢ (Lim 𝐵 ↔ (Ord 𝐵 ∧ 𝐵 ≠ ∅ ∧ 𝐵 = ∪ 𝐵)) | |
9 | 6, 7, 8 | 3bitr4g 314 | 1 ⊢ (𝐴 = 𝐵 → (Lim 𝐴 ↔ Lim 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1536 ≠ wne 2937 ∅c0 4338 ∪ cuni 4911 Ord word 6384 Lim wlim 6386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-v 3479 df-ss 3979 df-uni 4912 df-tr 5265 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-ord 6388 df-lim 6390 |
This theorem is referenced by: limuni2 6447 0ellim 6448 limuni3 7872 tfinds2 7884 dfom2 7888 limomss 7891 nnlim 7900 limom 7902 ssnlim 7906 onfununi 8379 tfr1a 8432 tz7.44lem1 8443 tz7.44-2 8445 tz7.44-3 8446 1ellim 8534 2ellim 8535 oeeulem 8637 limensuc 9192 elom3 9685 r1funlim 9803 rankxplim2 9917 rankxplim3 9918 rankxpsuc 9919 infxpenlem 10050 alephislim 10120 cflim2 10300 winalim 10732 rankcf 10814 gruina 10855 scutbdaybnd2lim 27876 rdgprc0 35774 dfrdg2 35776 dfrdg4 35932 limsucncmpi 36427 limsucncmp 36428 omlimcl2 43230 onexlimgt 43231 onov0suclim 43263 succlg 43317 dflim5 43318 nlim1NEW 43431 nlim2NEW 43432 nlim3 43433 nlim4 43434 dfsucon 43512 |
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