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Mirrors > Home > MPE Home > Th. List > isfin5-2 | Structured version Visualization version GIF version |
Description: Alternate definition of V-finite which emphasizes the idempotent behavior of V-infinite sets. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
Ref | Expression |
---|---|
isfin5-2 | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinV ↔ ¬ (𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 ⊔ 𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nne 2941 | . . . . 5 ⊢ (¬ 𝐴 ≠ ∅ ↔ 𝐴 = ∅) | |
2 | 1 | bicomi 224 | . . . 4 ⊢ (𝐴 = ∅ ↔ ¬ 𝐴 ≠ ∅) |
3 | 2 | a1i 11 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = ∅ ↔ ¬ 𝐴 ≠ ∅)) |
4 | djudoml 10222 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → 𝐴 ≼ (𝐴 ⊔ 𝐴)) | |
5 | 4 | anidms 566 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ (𝐴 ⊔ 𝐴)) |
6 | brsdom 9013 | . . . . 5 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) ↔ (𝐴 ≼ (𝐴 ⊔ 𝐴) ∧ ¬ 𝐴 ≈ (𝐴 ⊔ 𝐴))) | |
7 | 6 | baib 535 | . . . 4 ⊢ (𝐴 ≼ (𝐴 ⊔ 𝐴) → (𝐴 ≺ (𝐴 ⊔ 𝐴) ↔ ¬ 𝐴 ≈ (𝐴 ⊔ 𝐴))) |
8 | 5, 7 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≺ (𝐴 ⊔ 𝐴) ↔ ¬ 𝐴 ≈ (𝐴 ⊔ 𝐴))) |
9 | 3, 8 | orbi12d 918 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)) ↔ (¬ 𝐴 ≠ ∅ ∨ ¬ 𝐴 ≈ (𝐴 ⊔ 𝐴)))) |
10 | isfin5 10336 | . 2 ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴))) | |
11 | ianor 983 | . 2 ⊢ (¬ (𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 ⊔ 𝐴)) ↔ (¬ 𝐴 ≠ ∅ ∨ ¬ 𝐴 ≈ (𝐴 ⊔ 𝐴))) | |
12 | 9, 10, 11 | 3bitr4g 314 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinV ↔ ¬ (𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 ⊔ 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∅c0 4338 class class class wbr 5147 ≈ cen 8980 ≼ cdom 8981 ≺ csdm 8982 ⊔ cdju 9935 FinVcfin5 10319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ord 6388 df-on 6389 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-1st 8012 df-2nd 8013 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-dju 9938 df-fin5 10326 |
This theorem is referenced by: fin45 10429 |
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