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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 4572 | . . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | difeq1 4073 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∖ 𝑦) = (𝐴 ∖ 𝑦)) | |
3 | 2 | eleq1d 2822 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑦) ∈ Fin)) |
4 | 3 | orbi2d 914 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin) ↔ (𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin))) |
5 | 1, 4 | raleqbidv 3317 | . 2 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin) ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin))) |
6 | df-fin1a 10179 | . 2 ⊢ FinIa = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin)} | |
7 | 5, 6 | elab2g 3630 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIa ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ∀wral 3062 ∖ cdif 3905 𝒫 cpw 4558 Fincfn 8841 FinIacfin1a 10172 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3063 df-rab 3406 df-v 3445 df-dif 3911 df-in 3915 df-ss 3925 df-pw 4560 df-fin1a 10179 |
This theorem is referenced by: fin1ai 10187 fin11a 10277 enfin1ai 10278 |
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