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Theorem domfin4 10227

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 8903	. . 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 8944	. . . 4 ⊢ (𝐵 ≈ 𝑥 → 𝑥 ≈ 𝐵)
4	3	ad2antrl 729	. . 3 ⊢ (((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝑥 ≈ 𝐵)
5		ssfin4 10226	. . . 4 ⊢ ((𝐴 ∈ Fin^IV ∧ 𝑥 ⊆ 𝐴) → 𝑥 ∈ Fin^IV)
6	5	ad2ant2rl 750	. . 3 ⊢ (((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝑥 ∈ Fin^IV)
7		fin4en1 10225	. . 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 1937	1 ⊢ ((𝐴 ∈ Fin^IV ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin^IV)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∃wex 1781 ∈ wcel 2114 ⊆ wss 3890 class class class wbr 5086 ≈ cen 8884 ≼ cdom 8885 Fin^IVcfin4 10196

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-er 8637 df-en 8888 df-dom 8889 df-fin4 10203

This theorem is referenced by: infpssALT 10229