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Theorem domfin4 9345

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 8137	. . 3 ⊢ (𝐴 ∈ Fin^IV → (𝐵 ≼ 𝐴 ↔ ∃𝑥(𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)))
2	1	biimpa 502	. 2 ⊢ ((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) → ∃𝑥(𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴))
3		ensym 8172	. . . 4 ⊢ (𝐵 ≈ 𝑥 → 𝑥 ≈ 𝐵)
4	3	ad2antrl 766	. . 3 ⊢ (((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝑥 ≈ 𝐵)
5		ssfin4 9344	. . . 4 ⊢ ((𝐴 ∈ Fin^IV ∧ 𝑥 ⊆ 𝐴) → 𝑥 ∈ Fin^IV)
6	5	ad2ant2rl 802	. . 3 ⊢ (((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝑥 ∈ Fin^IV)
7		fin4en1 9343	. . 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 2012	1 ⊢ ((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin^IV)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∃wex 1853 ∈ wcel 2139 ⊆ wss 3715 class class class wbr 4804 ≈ cen 8120 ≼ cdom 8121 Fin^IVcfin4 9314

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-er 7913 df-en 8124 df-dom 8125 df-fin4 9321

This theorem is referenced by: infpssALT 9347