![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7855 | . . 3 ⊢ (𝐴 ∈ FinIV → (𝐵 ≼ 𝐴 ↔ ∃𝑥(𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) | |
2 | 1 | biimpa 500 | . 2 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ≼ 𝐴) → ∃𝑥(𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) |
3 | ensym 7891 | . . . 4 ⊢ (𝐵 ≈ 𝑥 → 𝑥 ≈ 𝐵) | |
4 | 3 | ad2antrl 760 | . . 3 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝑥 ≈ 𝐵) |
5 | ssfin4 9015 | . . . 4 ⊢ ((𝐴 ∈ FinIV ∧ 𝑥 ⊆ 𝐴) → 𝑥 ∈ FinIV) | |
6 | 5 | ad2ant2rl 781 | . . 3 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝑥 ∈ FinIV) |
7 | fin4en1 9014 | . . 3 ⊢ (𝑥 ≈ 𝐵 → (𝑥 ∈ FinIV → 𝐵 ∈ FinIV)) | |
8 | 4, 6, 7 | sylc 63 | . 2 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝐵 ∈ FinIV) |
9 | 2, 8 | exlimddv 1850 | 1 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ FinIV) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∃wex 1695 ∈ wcel 1977 ⊆ wss 3540 class class class wbr 4583 ≈ cen 7838 ≼ cdom 7839 FinIVcfin4 8985 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-er 7629 df-en 7842 df-dom 7843 df-fin4 8992 |
This theorem is referenced by: infpssALT 9018 |
Copyright terms: Public domain | W3C validator |