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Theorem domfin4 10351

Description: A set dominated by a Dedekind finite set is Dedekind finite. (Contributed by Mario Carneiro, 16-May-2015.)

**Assertion**
Ref	Expression
domfin4	⊢ ((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin^IV)

Proof of Theorem domfin4 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		domeng 9003	. . 3 ⊢ (𝐴 ∈ Fin^IV → (𝐵 ≼ 𝐴 ↔ ∃𝑥(𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)))
2	1	biimpa 476	. 2 ⊢ ((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) → ∃𝑥(𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴))
3		ensym 9043	. . . 4 ⊢ (𝐵 ≈ 𝑥 → 𝑥 ≈ 𝐵)
4	3	ad2antrl 728	. . 3 ⊢ (((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝑥 ≈ 𝐵)
5		ssfin4 10350	. . . 4 ⊢ ((𝐴 ∈ Fin^IV ∧ 𝑥 ⊆ 𝐴) → 𝑥 ∈ Fin^IV)
6	5	ad2ant2rl 749	. . 3 ⊢ (((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝑥 ∈ Fin^IV)
7		fin4en1 10349	. . 3 ⊢ (𝑥 ≈ 𝐵 → (𝑥 ∈ Fin^IV → 𝐵 ∈ Fin^IV))
8	4, 6, 7	sylc 65	. 2 ⊢ (((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝐵 ∈ Fin^IV)
9	2, 8	exlimddv 1935	1 ⊢ ((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin^IV)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∃wex 1779 ∈ wcel 2108 ⊆ wss 3951 class class class wbr 5143 ≈ cen 8982 ≼ cdom 8983 Fin^IVcfin4 10320

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-fin4 10327

This theorem is referenced by: infpssALT 10353