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 9713 | . . 3 ⊢ FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))} | |
2 | 1 | eleq2i 2906 | . 2 ⊢ (𝐴 ∈ FinV ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))}) |
3 | id 22 | . . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
4 | 0ex 5213 | . . . . 5 ⊢ ∅ ∈ V | |
5 | 3, 4 | eqeltrdi 2923 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 ∈ V) |
6 | relsdom 8518 | . . . . 5 ⊢ Rel ≺ | |
7 | 6 | brrelex1i 5610 | . . . 4 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → 𝐴 ∈ V) |
8 | 5, 7 | jaoi 853 | . . 3 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)) → 𝐴 ∈ V) |
9 | eqeq1 2827 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅)) | |
10 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
11 | djueq12 9335 | . . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 ⊔ 𝑥) = (𝐴 ⊔ 𝐴)) | |
12 | 11 | anidms 569 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊔ 𝑥) = (𝐴 ⊔ 𝐴)) |
13 | 10, 12 | breq12d 5081 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 ⊔ 𝑥) ↔ 𝐴 ≺ (𝐴 ⊔ 𝐴))) |
14 | 9, 13 | orbi12d 915 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥)) ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)))) |
15 | 8, 14 | elab3 3676 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))} ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴))) |
16 | 2, 15 | bitri 277 | 1 ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∨ wo 843 = wceq 1537 ∈ wcel 2114 {cab 2801 Vcvv 3496 ∅c0 4293 class class class wbr 5068 ≺ csdm 8510 ⊔ cdju 9329 FinVcfin5 9706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-dom 8513 df-sdom 8514 df-dju 9332 df-fin5 9713 |
This theorem is referenced by: isfin5-2 9815 fin56 9817 |
Copyright terms: Public domain | W3C validator |