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Mirrors > Home > MPE Home > Th. List > isfin1a | Structured version Visualization version GIF version |
Description: Definition of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
Ref | Expression |
---|---|
isfin1a | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIa ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pweq 4554 | . . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | difeq1 4091 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∖ 𝑦) = (𝐴 ∖ 𝑦)) | |
3 | 2 | eleq1d 2897 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑦) ∈ Fin)) |
4 | 3 | orbi2d 912 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin) ↔ (𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin))) |
5 | 1, 4 | raleqbidv 3401 | . 2 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin) ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin))) |
6 | df-fin1a 9706 | . 2 ⊢ FinIa = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin)} | |
7 | 5, 6 | elab2g 3667 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIa ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∖ cdif 3932 𝒫 cpw 4538 Fincfn 8508 FinIacfin1a 9699 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3143 df-rab 3147 df-dif 3938 df-in 3942 df-ss 3951 df-pw 4540 df-fin1a 9706 |
This theorem is referenced by: fin1ai 9714 fin11a 9804 enfin1ai 9805 |
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