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| Mirrors > Home > MPE Home > Th. List > isfin3 | Structured version Visualization version GIF version | ||
| Description: Definition of a III-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
| Ref | Expression |
|---|---|
| isfin3 | ⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-fin3 10328 | . . 3 ⊢ FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV} | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝐴 ∈ FinIII ↔ 𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}) |
| 3 | pwexr 7785 | . . 3 ⊢ (𝒫 𝐴 ∈ FinIV → 𝐴 ∈ V) | |
| 4 | pweq 4614 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
| 5 | 4 | eleq1d 2826 | . . 3 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ FinIV ↔ 𝒫 𝐴 ∈ FinIV)) |
| 6 | 3, 5 | elab3 3686 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV} ↔ 𝒫 𝐴 ∈ FinIV) |
| 7 | 2, 6 | bitri 275 | 1 ⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 {cab 2714 Vcvv 3480 𝒫 cpw 4600 FinIVcfin4 10320 FinIIIcfin3 10321 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-pw 4602 df-sn 4627 df-pr 4629 df-uni 4908 df-fin3 10328 |
| This theorem is referenced by: fin23lem41 10392 isfin32i 10405 fin34 10430 |
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