Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > isfin5 | Structured version Visualization version GIF version |
Description: Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
Ref | Expression |
---|---|
isfin5 | ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fin5 9699 | . . 3 ⊢ FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))} | |
2 | 1 | eleq2i 2901 | . 2 ⊢ (𝐴 ∈ FinV ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))}) |
3 | id 22 | . . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
4 | 0ex 5202 | . . . . 5 ⊢ ∅ ∈ V | |
5 | 3, 4 | syl6eqel 2918 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 ∈ V) |
6 | relsdom 8504 | . . . . 5 ⊢ Rel ≺ | |
7 | 6 | brrelex1i 5601 | . . . 4 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → 𝐴 ∈ V) |
8 | 5, 7 | jaoi 851 | . . 3 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)) → 𝐴 ∈ V) |
9 | eqeq1 2822 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅)) | |
10 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
11 | djueq12 9321 | . . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 ⊔ 𝑥) = (𝐴 ⊔ 𝐴)) | |
12 | 11 | anidms 567 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊔ 𝑥) = (𝐴 ⊔ 𝐴)) |
13 | 10, 12 | breq12d 5070 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 ⊔ 𝑥) ↔ 𝐴 ≺ (𝐴 ⊔ 𝐴))) |
14 | 9, 13 | orbi12d 912 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥)) ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)))) |
15 | 8, 14 | elab3 3671 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))} ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴))) |
16 | 2, 15 | bitri 276 | 1 ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∨ wo 841 = wceq 1528 ∈ wcel 2105 {cab 2796 Vcvv 3492 ∅c0 4288 class class class wbr 5057 ≺ csdm 8496 ⊔ cdju 9315 FinVcfin5 9692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-dom 8499 df-sdom 8500 df-dju 9318 df-fin5 9699 |
This theorem is referenced by: isfin5-2 9801 fin56 9803 |
Copyright terms: Public domain | W3C validator |