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Mirrors > Home > MPE Home > Th. List > fin45 | Structured version Visualization version GIF version |
Description: Every IV-finite set is V-finite: if we can pack two copies of the set into itself, we can certainly leave space. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 18-May-2015.) |
Ref | Expression |
---|---|
fin45 | ⊢ (𝐴 ∈ FinIV → 𝐴 ∈ FinV) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfin4-3 9452 | . . 3 ⊢ (𝐴 ∈ FinIV ↔ 𝐴 ≺ (𝐴 +𝑐 1o)) | |
2 | simpl 476 | . . . . . . . . 9 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 +𝑐 𝐴)) → 𝐴 ≠ ∅) | |
3 | relen 8227 | . . . . . . . . . . . 12 ⊢ Rel ≈ | |
4 | 3 | brrelex1i 5393 | . . . . . . . . . . 11 ⊢ (𝐴 ≈ (𝐴 +𝑐 𝐴) → 𝐴 ∈ V) |
5 | 4 | adantl 475 | . . . . . . . . . 10 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 +𝑐 𝐴)) → 𝐴 ∈ V) |
6 | 0sdomg 8358 | . . . . . . . . . 10 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
7 | 5, 6 | syl 17 | . . . . . . . . 9 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 +𝑐 𝐴)) → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
8 | 2, 7 | mpbird 249 | . . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 +𝑐 𝐴)) → ∅ ≺ 𝐴) |
9 | 0sdom1dom 8427 | . . . . . . . 8 ⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴) | |
10 | 8, 9 | sylib 210 | . . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 +𝑐 𝐴)) → 1o ≼ 𝐴) |
11 | cdadom2 9324 | . . . . . . 7 ⊢ (1o ≼ 𝐴 → (𝐴 +𝑐 1o) ≼ (𝐴 +𝑐 𝐴)) | |
12 | 10, 11 | syl 17 | . . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 +𝑐 𝐴)) → (𝐴 +𝑐 1o) ≼ (𝐴 +𝑐 𝐴)) |
13 | domen2 8372 | . . . . . . 7 ⊢ (𝐴 ≈ (𝐴 +𝑐 𝐴) → ((𝐴 +𝑐 1o) ≼ 𝐴 ↔ (𝐴 +𝑐 1o) ≼ (𝐴 +𝑐 𝐴))) | |
14 | 13 | adantl 475 | . . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 +𝑐 𝐴)) → ((𝐴 +𝑐 1o) ≼ 𝐴 ↔ (𝐴 +𝑐 1o) ≼ (𝐴 +𝑐 𝐴))) |
15 | 12, 14 | mpbird 249 | . . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 +𝑐 𝐴)) → (𝐴 +𝑐 1o) ≼ 𝐴) |
16 | domnsym 8355 | . . . . 5 ⊢ ((𝐴 +𝑐 1o) ≼ 𝐴 → ¬ 𝐴 ≺ (𝐴 +𝑐 1o)) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 +𝑐 𝐴)) → ¬ 𝐴 ≺ (𝐴 +𝑐 1o)) |
18 | 17 | con2i 137 | . . 3 ⊢ (𝐴 ≺ (𝐴 +𝑐 1o) → ¬ (𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 +𝑐 𝐴))) |
19 | 1, 18 | sylbi 209 | . 2 ⊢ (𝐴 ∈ FinIV → ¬ (𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 +𝑐 𝐴))) |
20 | isfin5-2 9528 | . 2 ⊢ (𝐴 ∈ FinIV → (𝐴 ∈ FinV ↔ ¬ (𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 +𝑐 𝐴)))) | |
21 | 19, 20 | mpbird 249 | 1 ⊢ (𝐴 ∈ FinIV → 𝐴 ∈ FinV) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2164 ≠ wne 2999 Vcvv 3414 ∅c0 4144 class class class wbr 4873 (class class class)co 6905 1oc1o 7819 ≈ cen 8219 ≼ cdom 8220 ≺ csdm 8221 +𝑐 ccda 9304 FinIVcfin4 9417 FinVcfin5 9419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-cda 9305 df-fin4 9424 df-fin5 9426 |
This theorem is referenced by: fin2so 33932 |
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