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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 9275 | . . 3 ⊢ FinVI = {𝑥 ∣ (𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥))} | |
2 | 1 | eleq2i 2819 | . 2 ⊢ (𝐴 ∈ FinVI ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥))}) |
3 | relsdom 8116 | . . . . 5 ⊢ Rel ≺ | |
4 | 3 | brrelexi 5303 | . . . 4 ⊢ (𝐴 ≺ 2𝑜 → 𝐴 ∈ V) |
5 | 3 | brrelexi 5303 | . . . 4 ⊢ (𝐴 ≺ (𝐴 × 𝐴) → 𝐴 ∈ V) |
6 | 4, 5 | jaoi 393 | . . 3 ⊢ ((𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴)) → 𝐴 ∈ V) |
7 | breq1 4795 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ 2𝑜 ↔ 𝐴 ≺ 2𝑜)) | |
8 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
9 | 8 | sqxpeqd 5286 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴)) |
10 | 8, 9 | breq12d 4805 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 × 𝑥) ↔ 𝐴 ≺ (𝐴 × 𝐴))) |
11 | 7, 10 | orbi12d 748 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥)) ↔ (𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴)))) |
12 | 6, 11 | elab3 3486 | . 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 382 = wceq 1620 ∈ wcel 2127 {cab 2734 Vcvv 3328 class class class wbr 4792 × cxp 5252 2𝑜c2o 7711 ≺ csdm 8108 FinVIcfin6 9268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pr 5043 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ral 3043 df-rex 3044 df-rab 3047 df-v 3330 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-nul 4047 df-if 4219 df-sn 4310 df-pr 4312 df-op 4316 df-br 4793 df-opab 4853 df-xp 5260 df-rel 5261 df-dom 8111 df-sdom 8112 df-fin6 9275 |
This theorem is referenced by: fin56 9378 fin67 9380 |
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