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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 4513 | . . . 4 ⊢ (𝑧 = 𝑥 → 𝒫 𝑧 = 𝒫 𝑥) | |
2 | 1 | pweqd 4516 | . . 3 ⊢ (𝑧 = 𝑥 → 𝒫 𝒫 𝑧 = 𝒫 𝒫 𝑥) |
3 | 2 | raleqdv 3364 | . 2 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝒫 𝒫 𝑧((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦) ↔ ∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦))) |
4 | pweq 4513 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
5 | 4 | pweqd 4516 | . . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝒫 𝑥 = 𝒫 𝒫 𝐴) |
6 | 5 | raleqdv 3364 | . 2 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦) ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦))) |
7 | df-fin2 9697 | . 2 ⊢ FinII = {𝑧 ∣ ∀𝑦 ∈ 𝒫 𝒫 𝑧((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦)} | |
8 | 3, 6, 7 | elab2gw 3613 | 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∅c0 4243 𝒫 cpw 4497 ∪ cuni 4800 Or wor 5437 [⊊] crpss 7428 FinIIcfin2 9690 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-v 3443 df-in 3888 df-ss 3898 df-pw 4499 df-fin2 9697 |
This theorem is referenced by: fin2i 9706 isfin2-2 9730 ssfin2 9731 enfin2i 9732 fin12 9824 fin1a2s 9825 |
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