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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 10326 | . . 3 ⊢ FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV} | |
2 | 1 | eleq2i 2831 | . 2 ⊢ (𝐴 ∈ FinIII ↔ 𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}) |
3 | pwexr 7784 | . . 3 ⊢ (𝒫 𝐴 ∈ FinIV → 𝐴 ∈ V) | |
4 | pweq 4619 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
5 | 4 | eleq1d 2824 | . . 3 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ FinIV ↔ 𝒫 𝐴 ∈ FinIV)) |
6 | 3, 5 | elab3 3689 | . 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 1537 ∈ wcel 2106 {cab 2712 Vcvv 3478 𝒫 cpw 4605 FinIVcfin4 10318 FinIIIcfin3 10319 |
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 ax-sep 5302 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-ss 3980 df-pw 4607 df-sn 4632 df-pr 4634 df-uni 4913 df-fin3 10326 |
This theorem is referenced by: fin23lem41 10390 isfin32i 10403 fin34 10428 |
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