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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 9112 | . . 3 ⊢ FinVI = {𝑥 ∣ (𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥))} | |
2 | 1 | eleq2i 2693 | . 2 ⊢ (𝐴 ∈ FinVI ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥))}) |
3 | relsdom 7962 | . . . . 5 ⊢ Rel ≺ | |
4 | 3 | brrelexi 5158 | . . . 4 ⊢ (𝐴 ≺ 2𝑜 → 𝐴 ∈ V) |
5 | 3 | brrelexi 5158 | . . . 4 ⊢ (𝐴 ≺ (𝐴 × 𝐴) → 𝐴 ∈ V) |
6 | 4, 5 | jaoi 394 | . . 3 ⊢ ((𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴)) → 𝐴 ∈ V) |
7 | breq1 4656 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ 2𝑜 ↔ 𝐴 ≺ 2𝑜)) | |
8 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
9 | 8 | sqxpeqd 5141 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴)) |
10 | 8, 9 | breq12d 4666 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 × 𝑥) ↔ 𝐴 ≺ (𝐴 × 𝐴))) |
11 | 7, 10 | orbi12d 746 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥)) ↔ (𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴)))) |
12 | 6, 11 | elab3 3358 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥))} ↔ (𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
13 | 2, 12 | bitri 264 | 1 ⊢ (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∨ wo 383 = wceq 1483 ∈ wcel 1990 {cab 2608 Vcvv 3200 class class class wbr 4653 × cxp 5112 2𝑜c2o 7554 ≺ csdm 7954 FinVIcfin6 9105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-dom 7957 df-sdom 7958 df-fin6 9112 |
This theorem is referenced by: fin56 9215 fin67 9217 |
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