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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 +𝑐 𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nne 2975 | . . . . 5 ⊢ (¬ 𝐴 ≠ ∅ ↔ 𝐴 = ∅) | |
2 | 1 | bicomi 216 | . . . 4 ⊢ (𝐴 = ∅ ↔ ¬ 𝐴 ≠ ∅) |
3 | 2 | a1i 11 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = ∅ ↔ ¬ 𝐴 ≠ ∅)) |
4 | cdadom3 9298 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → 𝐴 ≼ (𝐴 +𝑐 𝐴)) | |
5 | 4 | anidms 563 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ (𝐴 +𝑐 𝐴)) |
6 | brsdom 8218 | . . . . 5 ⊢ (𝐴 ≺ (𝐴 +𝑐 𝐴) ↔ (𝐴 ≼ (𝐴 +𝑐 𝐴) ∧ ¬ 𝐴 ≈ (𝐴 +𝑐 𝐴))) | |
7 | 6 | baib 532 | . . . 4 ⊢ (𝐴 ≼ (𝐴 +𝑐 𝐴) → (𝐴 ≺ (𝐴 +𝑐 𝐴) ↔ ¬ 𝐴 ≈ (𝐴 +𝑐 𝐴))) |
8 | 5, 7 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≺ (𝐴 +𝑐 𝐴) ↔ ¬ 𝐴 ≈ (𝐴 +𝑐 𝐴))) |
9 | 3, 8 | orbi12d 943 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴)) ↔ (¬ 𝐴 ≠ ∅ ∨ ¬ 𝐴 ≈ (𝐴 +𝑐 𝐴)))) |
10 | isfin5 9409 | . 2 ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴))) | |
11 | ianor 1005 | . 2 ⊢ (¬ (𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 +𝑐 𝐴)) ↔ (¬ 𝐴 ≠ ∅ ∨ ¬ 𝐴 ≈ (𝐴 +𝑐 𝐴))) | |
12 | 9, 10, 11 | 3bitr4g 306 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinV ↔ ¬ (𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 +𝑐 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 385 ∨ wo 874 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 ∅c0 4115 class class class wbr 4843 (class class class)co 6878 ≈ cen 8192 ≼ cdom 8193 ≺ csdm 8194 +𝑐 ccda 9277 FinVcfin5 9392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-ord 5944 df-on 5945 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-1o 7799 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-cda 9278 df-fin5 9399 |
This theorem is referenced by: fin45 9502 |
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