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Mirrors > Home > MPE Home > Th. List > isfin6 | Structured version Visualization version GIF version |
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 9400 | . . 3 ⊢ FinVI = {𝑥 ∣ (𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥))} | |
2 | 1 | eleq2i 2870 | . 2 ⊢ (𝐴 ∈ FinVI ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥))}) |
3 | relsdom 8202 | . . . . 5 ⊢ Rel ≺ | |
4 | 3 | brrelex1i 5363 | . . . 4 ⊢ (𝐴 ≺ 2𝑜 → 𝐴 ∈ V) |
5 | 3 | brrelex1i 5363 | . . . 4 ⊢ (𝐴 ≺ (𝐴 × 𝐴) → 𝐴 ∈ V) |
6 | 4, 5 | jaoi 884 | . . 3 ⊢ ((𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴)) → 𝐴 ∈ V) |
7 | breq1 4846 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ 2𝑜 ↔ 𝐴 ≺ 2𝑜)) | |
8 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
9 | 8 | sqxpeqd 5344 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴)) |
10 | 8, 9 | breq12d 4856 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 × 𝑥) ↔ 𝐴 ≺ (𝐴 × 𝐴))) |
11 | 7, 10 | orbi12d 943 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥)) ↔ (𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴)))) |
12 | 6, 11 | elab3 3550 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥))} ↔ (𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
13 | 2, 12 | bitri 267 | 1 ⊢ (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∨ wo 874 = wceq 1653 ∈ wcel 2157 {cab 2785 Vcvv 3385 class class class wbr 4843 × cxp 5310 2𝑜c2o 7793 ≺ csdm 8194 FinVIcfin6 9393 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-xp 5318 df-rel 5319 df-dom 8197 df-sdom 8198 df-fin6 9400 |
This theorem is referenced by: fin56 9503 fin67 9505 |
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