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| Mirrors > Home > MPE Home > Th. List > dflim2 | Structured version Visualization version GIF version | ||
| Description: An alternate definition of a limit ordinal. (Contributed by NM, 4-Nov-2004.) | 
| Ref | Expression | 
|---|---|
| dflim2 | ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-lim 6388 | . 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | |
| 2 | ord0eln0 6438 | . . . . 5 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
| 3 | 2 | anbi1d 631 | . . . 4 ⊢ (Ord 𝐴 → ((∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴) ↔ (𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))) | 
| 4 | 3 | pm5.32i 574 | . . 3 ⊢ ((Ord 𝐴 ∧ (∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴)) ↔ (Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))) | 
| 5 | 3anass 1094 | . . 3 ⊢ ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴) ↔ (Ord 𝐴 ∧ (∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴))) | |
| 6 | 3anass 1094 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) ↔ (Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))) | |
| 7 | 4, 5, 6 | 3bitr4i 303 | . 2 ⊢ ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴) ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | 
| 8 | 1, 7 | bitr4i 278 | 1 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∅c0 4332 ∪ cuni 4906 Ord word 6382 Lim wlim 6384 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-lim 6388 | 
| This theorem is referenced by: nlim0 6442 dflim4 7870 nlimsuc 43459 | 
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