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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 9704 | . . 3 ⊢ FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV} | |
2 | 1 | eleq2i 2904 | . 2 ⊢ (𝐴 ∈ FinIII ↔ 𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}) |
3 | pwexr 7481 | . . 3 ⊢ (𝒫 𝐴 ∈ FinIV → 𝐴 ∈ V) | |
4 | pweq 4541 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
5 | 4 | eleq1d 2897 | . . 3 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ FinIV ↔ 𝒫 𝐴 ∈ FinIV)) |
6 | 3, 5 | elab3 3673 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV} ↔ 𝒫 𝐴 ∈ FinIV) |
7 | 2, 6 | bitri 277 | 1 ⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ∈ wcel 2110 {cab 2799 𝒫 cpw 4538 FinIVcfin4 9696 FinIIIcfin3 9697 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-pw 4540 df-sn 4561 df-pr 4563 df-uni 4832 df-fin3 9704 |
This theorem is referenced by: fin23lem41 9768 isfin32i 9781 fin34 9806 |
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