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Theorem domfin4 9586

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 8378	. . 3 ⊢ (𝐴 ∈ Fin^IV → (𝐵 ≼ 𝐴 ↔ ∃𝑥(𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)))
2	1	biimpa 477	. 2 ⊢ ((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) → ∃𝑥(𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴))
3		ensym 8413	. . . 4 ⊢ (𝐵 ≈ 𝑥 → 𝑥 ≈ 𝐵)
4	3	ad2antrl 724	. . 3 ⊢ (((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝑥 ≈ 𝐵)
5		ssfin4 9585	. . . 4 ⊢ ((𝐴 ∈ Fin^IV ∧ 𝑥 ⊆ 𝐴) → 𝑥 ∈ Fin^IV)
6	5	ad2ant2rl 745	. . 3 ⊢ (((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝑥 ∈ Fin^IV)
7		fin4en1 9584	. . 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 1917	1 ⊢ ((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin^IV)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∃wex 1765 ∈ wcel 2083 ⊆ wss 3865 class class class wbr 4968 ≈ cen 8361 ≼ cdom 8362 Fin^IVcfin4 9555

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-er 8146 df-en 8365 df-dom 8366 df-fin4 9562

This theorem is referenced by: infpssALT 9588