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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 8823 | . . 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 8864 | . . . 4 ⊢ (𝐵 ≈ 𝑥 → 𝑥 ≈ 𝐵) | |
4 | 3 | ad2antrl 725 | . . 3 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝑥 ≈ 𝐵) |
5 | ssfin4 10167 | . . . 4 ⊢ ((𝐴 ∈ FinIV ∧ 𝑥 ⊆ 𝐴) → 𝑥 ∈ FinIV) | |
6 | 5 | ad2ant2rl 746 | . . 3 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝑥 ∈ FinIV) |
7 | fin4en1 10166 | . . 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 1937 | 1 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ FinIV) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∃wex 1780 ∈ wcel 2105 ⊆ wss 3898 class class class wbr 5092 ≈ cen 8801 ≼ cdom 8802 FinIVcfin4 10137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-er 8569 df-en 8805 df-dom 8806 df-fin4 10144 |
This theorem is referenced by: infpssALT 10170 |
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