fin11a - Metamath Proof Explorer

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Theorem fin11a 10343

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 4573	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
2		ssfi 9143	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ⊆ 𝐴) → 𝑥 ∈ Fin)
3	1, 2	sylan2 593	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ Fin)
4	3	orcd 873	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin))
5	4	ralrimiva 3126	. 2 ⊢ (𝐴 ∈ Fin → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin))
6		isfin1a 10252	. 2 ⊢ (𝐴 ∈ Fin → (𝐴 ∈ Fin^Ia ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin)))
7	5, 6	mpbird 257	1 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ Fin^Ia)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 ∈ wcel 2109 ∀wral 3045 ∖ cdif 3914 ⊆ wss 3917 𝒫 cpw 4566 Fincfn 8921 Fin^Iacfin1a 10238

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-om 7846 df-1o 8437 df-en 8922 df-fin 8925 df-fin1a 10245

This theorem is referenced by: (None)