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| Description: Definition of a VI-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) | 
| Ref | Expression | 
|---|---|
| isfin6 | ⊢ (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-fin6 10331 | . . 3 ⊢ FinVI = {𝑥 ∣ (𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥))} | |
| 2 | 1 | eleq2i 2832 | . 2 ⊢ (𝐴 ∈ FinVI ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥))}) | 
| 3 | relsdom 8993 | . . . . 5 ⊢ Rel ≺ | |
| 4 | 3 | brrelex1i 5740 | . . . 4 ⊢ (𝐴 ≺ 2o → 𝐴 ∈ V) | 
| 5 | 3 | brrelex1i 5740 | . . . 4 ⊢ (𝐴 ≺ (𝐴 × 𝐴) → 𝐴 ∈ V) | 
| 6 | 4, 5 | jaoi 857 | . . 3 ⊢ ((𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴)) → 𝐴 ∈ V) | 
| 7 | breq1 5145 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ 2o ↔ 𝐴 ≺ 2o)) | |
| 8 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
| 9 | 8 | sqxpeqd 5716 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴)) | 
| 10 | 8, 9 | breq12d 5155 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 × 𝑥) ↔ 𝐴 ≺ (𝐴 × 𝐴))) | 
| 11 | 7, 10 | orbi12d 918 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥)) ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴)))) | 
| 12 | 6, 11 | elab3 3685 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥))} ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) | 
| 13 | 2, 12 | bitri 275 | 1 ⊢ (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∨ wo 847 = wceq 1539 ∈ wcel 2107 {cab 2713 Vcvv 3479 class class class wbr 5142 × cxp 5682 2oc2o 8501 ≺ csdm 8985 FinVIcfin6 10324 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-dom 8988 df-sdom 8989 df-fin6 10331 | 
| This theorem is referenced by: fin56 10434 fin67 10436 | 
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