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Mirrors > Home > MPE Home > Th. List > domfin4 | Structured version Visualization version GIF version |
Description: A set dominated by a Dedekind finite set is Dedekind finite. (Contributed by Mario Carneiro, 16-May-2015.) |
Ref | Expression |
---|---|
domfin4 | ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ FinIV) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domeng 8901 | . . 3 ⊢ (𝐴 ∈ FinIV → (𝐵 ≼ 𝐴 ↔ ∃𝑥(𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) | |
2 | 1 | biimpa 477 | . 2 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ≼ 𝐴) → ∃𝑥(𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) |
3 | ensym 8942 | . . . 4 ⊢ (𝐵 ≈ 𝑥 → 𝑥 ≈ 𝐵) | |
4 | 3 | ad2antrl 726 | . . 3 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝑥 ≈ 𝐵) |
5 | ssfin4 10245 | . . . 4 ⊢ ((𝐴 ∈ FinIV ∧ 𝑥 ⊆ 𝐴) → 𝑥 ∈ FinIV) | |
6 | 5 | ad2ant2rl 747 | . . 3 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝑥 ∈ FinIV) |
7 | fin4en1 10244 | . . 3 ⊢ (𝑥 ≈ 𝐵 → (𝑥 ∈ FinIV → 𝐵 ∈ FinIV)) | |
8 | 4, 6, 7 | sylc 65 | . 2 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝐵 ∈ FinIV) |
9 | 2, 8 | exlimddv 1938 | 1 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ FinIV) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∃wex 1781 ∈ wcel 2106 ⊆ wss 3910 class class class wbr 5105 ≈ cen 8879 ≼ cdom 8880 FinIVcfin4 10215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-er 8647 df-en 8883 df-dom 8884 df-fin4 10222 |
This theorem is referenced by: infpssALT 10248 |
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