fin11a - Metamath Proof Explorer

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Theorem fin11a 9088

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 4117	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
2		ssfi 8065	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ⊆ 𝐴) → 𝑥 ∈ Fin)
3	1, 2	sylan2 490	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ Fin)
4	3	orcd 406	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin))
5	4	ralrimiva 2949	. 2 ⊢ (𝐴 ∈ Fin → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin))
6		isfin1a 8997	. 2 ⊢ (𝐴 ∈ Fin → (𝐴 ∈ Fin^Ia ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin)))
7	5, 6	mpbird 246	1 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ Fin^Ia)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 ∖ cdif 3537 ⊆ wss 3540 𝒫 cpw 4108 Fincfn 7841 Fin^Iacfin1a 8983

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-om 6958 df-er 7629 df-en 7842 df-fin 7845 df-fin1a 8990

This theorem is referenced by: (None)