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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 10282 | . . 3 ⊢ FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV} | |
2 | 1 | eleq2i 2819 | . 2 ⊢ (𝐴 ∈ FinIII ↔ 𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}) |
3 | pwexr 7748 | . . 3 ⊢ (𝒫 𝐴 ∈ FinIV → 𝐴 ∈ V) | |
4 | pweq 4611 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
5 | 4 | eleq1d 2812 | . . 3 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ FinIV ↔ 𝒫 𝐴 ∈ FinIV)) |
6 | 3, 5 | elab3 3671 | . 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 205 = wceq 1533 ∈ wcel 2098 {cab 2703 Vcvv 3468 𝒫 cpw 4597 FinIVcfin4 10274 FinIIIcfin3 10275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-un 3948 df-in 3950 df-ss 3960 df-pw 4599 df-sn 4624 df-pr 4626 df-uni 4903 df-fin3 10282 |
This theorem is referenced by: fin23lem41 10346 isfin32i 10359 fin34 10384 |
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