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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 | simpl 486 | . . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 ⊔ 𝐴)) → 𝐴 ≠ ∅) | |
2 | relen 8562 | . . . . . . . . . . 11 ⊢ Rel ≈ | |
3 | 2 | brrelex1i 5579 | . . . . . . . . . 10 ⊢ (𝐴 ≈ (𝐴 ⊔ 𝐴) → 𝐴 ∈ V) |
4 | 3 | adantl 485 | . . . . . . . . 9 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 ⊔ 𝐴)) → 𝐴 ∈ V) |
5 | 0sdomg 8698 | . . . . . . . . 9 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | |
6 | 4, 5 | syl 17 | . . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 ⊔ 𝐴)) → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) |
7 | 1, 6 | mpbird 260 | . . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 ⊔ 𝐴)) → ∅ ≺ 𝐴) |
8 | 0sdom1dom 8797 | . . . . . . 7 ⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴) | |
9 | 7, 8 | sylib 221 | . . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 ⊔ 𝐴)) → 1o ≼ 𝐴) |
10 | djudom2 9685 | . . . . . 6 ⊢ ((1o ≼ 𝐴 ∧ 𝐴 ∈ V) → (𝐴 ⊔ 1o) ≼ (𝐴 ⊔ 𝐴)) | |
11 | 9, 4, 10 | syl2anc 587 | . . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 ⊔ 𝐴)) → (𝐴 ⊔ 1o) ≼ (𝐴 ⊔ 𝐴)) |
12 | domen2 8712 | . . . . . 6 ⊢ (𝐴 ≈ (𝐴 ⊔ 𝐴) → ((𝐴 ⊔ 1o) ≼ 𝐴 ↔ (𝐴 ⊔ 1o) ≼ (𝐴 ⊔ 𝐴))) | |
13 | 12 | adantl 485 | . . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 ⊔ 𝐴)) → ((𝐴 ⊔ 1o) ≼ 𝐴 ↔ (𝐴 ⊔ 1o) ≼ (𝐴 ⊔ 𝐴))) |
14 | 11, 13 | mpbird 260 | . . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 ⊔ 𝐴)) → (𝐴 ⊔ 1o) ≼ 𝐴) |
15 | domnsym 8695 | . . . 4 ⊢ ((𝐴 ⊔ 1o) ≼ 𝐴 → ¬ 𝐴 ≺ (𝐴 ⊔ 1o)) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 ⊔ 𝐴)) → ¬ 𝐴 ≺ (𝐴 ⊔ 1o)) |
17 | isfin4p1 9817 | . . . 4 ⊢ (𝐴 ∈ FinIV ↔ 𝐴 ≺ (𝐴 ⊔ 1o)) | |
18 | 17 | biimpi 219 | . . 3 ⊢ (𝐴 ∈ FinIV → 𝐴 ≺ (𝐴 ⊔ 1o)) |
19 | 16, 18 | nsyl3 140 | . 2 ⊢ (𝐴 ∈ FinIV → ¬ (𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 ⊔ 𝐴))) |
20 | isfin5-2 9893 | . 2 ⊢ (𝐴 ∈ FinIV → (𝐴 ∈ FinV ↔ ¬ (𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 ⊔ 𝐴)))) | |
21 | 19, 20 | mpbird 260 | 1 ⊢ (𝐴 ∈ FinIV → 𝐴 ∈ FinV) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2114 ≠ wne 2934 Vcvv 3398 ∅c0 4211 class class class wbr 5030 1oc1o 8126 ≈ cen 8554 ≼ cdom 8555 ≺ csdm 8556 ⊔ cdju 9402 FinIVcfin4 9782 FinVcfin5 9784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-om 7602 df-1st 7716 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-er 8322 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-dju 9405 df-fin4 9789 df-fin5 9791 |
This theorem is referenced by: fin2so 35409 |
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