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Theorem domfin4 10428

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 9027	. . 3 ⊢ (𝐴 ∈ Fin^IV → (𝐵 ≼ 𝐴 ↔ ∃𝑥(𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)))
2	1	biimpa 482	. 2 ⊢ ((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) → ∃𝑥(𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴))
3		ensym 9068	. . . 4 ⊢ (𝐵 ≈ 𝑥 → 𝑥 ≈ 𝐵)
4	3	ad2antrl 741	. . 3 ⊢ (((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝑥 ≈ 𝐵)
5		ssfin4 10427	. . . 4 ⊢ ((𝐴 ∈ Fin^IV ∧ 𝑥 ⊆ 𝐴) → 𝑥 ∈ Fin^IV)
6	5	ad2ant2rl 762	. . 3 ⊢ (((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝑥 ∈ Fin^IV)
7		fin4en1 10426	. . 3 ⊢ (𝑥 ≈ 𝐵 → (𝑥 ∈ Fin^IV → 𝐵 ∈ Fin^IV))
8	4, 6, 7	sylc 66	. 2 ⊢ (((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝐵 ∈ Fin^IV)
9	2, 8	exlimddv 1970	1 ⊢ ((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin^IV)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∃wex 1812 ∈ wcel 2147 ⊆ wss 3900 class class class wbr 5108 ≈ cen 9008 ≼ cdom 9009 Fin^IVcfin4 10397

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-er 8755 df-en 9012 df-dom 9013 df-fin4 10404

This theorem is used by: infpssALT 10430

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