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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 10287 | . . 3 ⊢ FinVI = {𝑥 ∣ (𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥))} | |
2 | 1 | eleq2i 2823 | . 2 ⊢ (𝐴 ∈ FinVI ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥))}) |
3 | relsdom 8948 | . . . . 5 ⊢ Rel ≺ | |
4 | 3 | brrelex1i 5731 | . . . 4 ⊢ (𝐴 ≺ 2o → 𝐴 ∈ V) |
5 | 3 | brrelex1i 5731 | . . . 4 ⊢ (𝐴 ≺ (𝐴 × 𝐴) → 𝐴 ∈ V) |
6 | 4, 5 | jaoi 853 | . . 3 ⊢ ((𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴)) → 𝐴 ∈ V) |
7 | breq1 5150 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ 2o ↔ 𝐴 ≺ 2o)) | |
8 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
9 | 8 | sqxpeqd 5707 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴)) |
10 | 8, 9 | breq12d 5160 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 × 𝑥) ↔ 𝐴 ≺ (𝐴 × 𝐴))) |
11 | 7, 10 | orbi12d 915 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥)) ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴)))) |
12 | 6, 11 | elab3 3675 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥))} ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
13 | 2, 12 | bitri 274 | 1 ⊢ (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 843 = wceq 1539 ∈ wcel 2104 {cab 2707 Vcvv 3472 class class class wbr 5147 × cxp 5673 2oc2o 8462 ≺ csdm 8940 FinVIcfin6 10280 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 df-dom 8943 df-sdom 8944 df-fin6 10287 |
This theorem is referenced by: fin56 10390 fin67 10392 |
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