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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 6320 | . 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) | |
2 | ord0eln0 6370 | . . . . 5 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
3 | 2 | anbi1d 630 | . . . 4 ⊢ (Ord 𝐴 → ((∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴) ↔ (𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))) |
4 | 3 | pm5.32i 575 | . . 3 ⊢ ((Ord 𝐴 ∧ (∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴)) ↔ (Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))) |
5 | 3anass 1095 | . . 3 ⊢ ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴) ↔ (Ord 𝐴 ∧ (∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴))) | |
6 | 3anass 1095 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴) ↔ (Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))) | |
7 | 4, 5, 6 | 3bitr4i 302 | . 2 ⊢ ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴) ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) |
8 | 1, 7 | bitr4i 277 | 1 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∅c0 4280 ∪ cuni 4863 Ord word 6314 Lim wlim 6316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-tr 5221 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-ord 6318 df-lim 6320 |
This theorem is referenced by: nlim0 6374 dflim4 7776 nlimsuc 41618 |
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