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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 2987 | . . . . 5 ⊢ (¬ 𝐴 ≠ ∅ ↔ 𝐴 = ∅) | |
2 | 1 | bicomi 225 | . . . 4 ⊢ (𝐴 = ∅ ↔ ¬ 𝐴 ≠ ∅) |
3 | 2 | a1i 11 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = ∅ ↔ ¬ 𝐴 ≠ ∅)) |
4 | djudoml 9459 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → 𝐴 ≼ (𝐴 ⊔ 𝐴)) | |
5 | 4 | anidms 567 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ (𝐴 ⊔ 𝐴)) |
6 | brsdom 8383 | . . . . 5 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) ↔ (𝐴 ≼ (𝐴 ⊔ 𝐴) ∧ ¬ 𝐴 ≈ (𝐴 ⊔ 𝐴))) | |
7 | 6 | baib 536 | . . . 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 913 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)) ↔ (¬ 𝐴 ≠ ∅ ∨ ¬ 𝐴 ≈ (𝐴 ⊔ 𝐴)))) |
10 | isfin5 9570 | . 2 ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴))) | |
11 | ianor 976 | . 2 ⊢ (¬ (𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 ⊔ 𝐴)) ↔ (¬ 𝐴 ≠ ∅ ∨ ¬ 𝐴 ≈ (𝐴 ⊔ 𝐴))) | |
12 | 9, 10, 11 | 3bitr4g 315 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinV ↔ ¬ (𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 ⊔ 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 842 = wceq 1522 ∈ wcel 2080 ≠ wne 2983 ∅c0 4213 class class class wbr 4964 ≈ cen 8357 ≼ cdom 8358 ≺ csdm 8359 ⊔ cdju 9176 FinVcfin5 9553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-ral 3109 df-rex 3110 df-rab 3113 df-v 3438 df-sbc 3708 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-int 4785 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-ord 6072 df-on 6073 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-1st 7548 df-2nd 7549 df-1o 7956 df-er 8142 df-en 8361 df-dom 8362 df-sdom 8363 df-dju 9179 df-fin5 9560 |
This theorem is referenced by: fin45 9663 |
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