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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 9699 | . . 3 ⊢ FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV} | |
2 | 1 | eleq2i 2881 | . 2 ⊢ (𝐴 ∈ FinIII ↔ 𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}) |
3 | pwexr 7467 | . . 3 ⊢ (𝒫 𝐴 ∈ FinIV → 𝐴 ∈ V) | |
4 | pweq 4513 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
5 | 4 | eleq1d 2874 | . . 3 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ FinIV ↔ 𝒫 𝐴 ∈ FinIV)) |
6 | 3, 5 | elab3 3622 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV} ↔ 𝒫 𝐴 ∈ FinIV) |
7 | 2, 6 | bitri 278 | 1 ⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 {cab 2776 𝒫 cpw 4497 FinIVcfin4 9691 FinIIIcfin3 9692 |
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-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-pw 4499 df-sn 4526 df-pr 4528 df-uni 4801 df-fin3 9699 |
This theorem is referenced by: fin23lem41 9763 isfin32i 9776 fin34 9801 |
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