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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 4541 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | 1 | pweqd 4543 | . . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝒫 𝑥 = 𝒫 𝒫 𝐴) |
3 | 2 | raleqdv 3415 | . 2 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦) ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦))) |
4 | df-fin2 9702 | . 2 ⊢ FinII = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦)} | |
5 | 3, 4 | elab2g 3667 | 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 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∅c0 4290 𝒫 cpw 4538 ∪ cuni 4831 Or wor 5467 [⊊] crpss 7442 FinIIcfin2 9695 |
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 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-in 3942 df-ss 3951 df-pw 4540 df-fin2 9702 |
This theorem is referenced by: fin2i 9711 isfin2-2 9735 ssfin2 9736 enfin2i 9737 fin12 9829 fin1a2s 9830 |
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