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Mirrors > Home > MPE Home > Th. List > isfin2 | Structured version Visualization version GIF version |
Description: Definition of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
Ref | Expression |
---|---|
isfin2 | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pweq 4619 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | 1 | pweqd 4622 | . . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝒫 𝑥 = 𝒫 𝒫 𝐴) |
3 | 2 | raleqdv 3324 | . 2 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦) ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦))) |
4 | df-fin2 10324 | . 2 ⊢ FinII = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦)} | |
5 | 3, 4 | elab2g 3683 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∅c0 4339 𝒫 cpw 4605 ∪ cuni 4912 Or wor 5596 [⊊] crpss 7741 FinIIcfin2 10317 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-v 3480 df-ss 3980 df-pw 4607 df-fin2 10324 |
This theorem is referenced by: fin2i 10333 isfin2-2 10357 ssfin2 10358 enfin2i 10359 fin12 10451 fin1a2s 10452 |
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