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| Description: Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) | 
| Ref | Expression | 
|---|---|
| isfin5 | ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-fin5 10329 | . . 3 ⊢ FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))} | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝐴 ∈ FinV ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))}) | 
| 3 | id 22 | . . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
| 4 | 0ex 5307 | . . . . 5 ⊢ ∅ ∈ V | |
| 5 | 3, 4 | eqeltrdi 2849 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 ∈ V) | 
| 6 | relsdom 8992 | . . . . 5 ⊢ Rel ≺ | |
| 7 | 6 | brrelex1i 5741 | . . . 4 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → 𝐴 ∈ V) | 
| 8 | 5, 7 | jaoi 858 | . . 3 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)) → 𝐴 ∈ V) | 
| 9 | eqeq1 2741 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅)) | |
| 10 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
| 11 | djueq12 9944 | . . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 ⊔ 𝑥) = (𝐴 ⊔ 𝐴)) | |
| 12 | 11 | anidms 566 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊔ 𝑥) = (𝐴 ⊔ 𝐴)) | 
| 13 | 10, 12 | breq12d 5156 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 ⊔ 𝑥) ↔ 𝐴 ≺ (𝐴 ⊔ 𝐴))) | 
| 14 | 9, 13 | orbi12d 919 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥)) ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)))) | 
| 15 | 8, 14 | elab3 3686 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))} ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴))) | 
| 16 | 2, 15 | bitri 275 | 1 ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∨ wo 848 = wceq 1540 ∈ wcel 2108 {cab 2714 Vcvv 3480 ∅c0 4333 class class class wbr 5143 ≺ csdm 8984 ⊔ cdju 9938 FinVcfin5 10322 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-dom 8987 df-sdom 8988 df-dju 9941 df-fin5 10329 | 
| This theorem is referenced by: isfin5-2 10431 fin56 10433 | 
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