fin11a - Metamath Proof Explorer

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Theorem fin11a 10139

Description: Every I-finite set is Ia-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)

**Assertion**
Ref	Expression
fin11a	⊢ (𝐴 ∈ Fin → 𝐴 ∈ Fin^Ia)

Proof of Theorem fin11a Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpwi 4542	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
2		ssfi 8956	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ⊆ 𝐴) → 𝑥 ∈ Fin)
3	1, 2	sylan2 593	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ Fin)
4	3	orcd 870	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin))
5	4	ralrimiva 3103	. 2 ⊢ (𝐴 ∈ Fin → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin))
6		isfin1a 10048	. 2 ⊢ (𝐴 ∈ Fin → (𝐴 ∈ Fin^Ia ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin)))
7	5, 6	mpbird 256	1 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ Fin^Ia)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ wo 844 ∈ wcel 2106 ∀wral 3064 ∖ cdif 3884 ⊆ wss 3887 𝒫 cpw 4533 Fincfn 8733 Fin^Iacfin1a 10034

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-om 7713 df-1o 8297 df-en 8734 df-fin 8737 df-fin1a 10041

This theorem is referenced by: (None)