fin1a2 - Metamath Proof Explorer

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Theorem fin1a2 10306

Description: Every Ia-finite set is II-finite. Theorem 1 of [Levy58], p. 3. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)

**Assertion**
Ref	Expression
fin1a2	⊢ (𝐴 ∈ Fin^Ia → 𝐴 ∈ Fin^II)

Proof of Theorem fin1a2 Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpwi 4557	. . . 4 ⊢ (𝑏 ∈ 𝒫 𝐴 → 𝑏 ⊆ 𝐴)
2		fin1ai 10184	. . . . 5 ⊢ ((𝐴 ∈ Fin^Ia ∧ 𝑏 ⊆ 𝐴) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin))
3		fin12 10304	. . . . . 6 ⊢ ((𝐴 ∖ 𝑏) ∈ Fin → (𝐴 ∖ 𝑏) ∈ Fin^II)
4	3	orim2i 910	. . . . 5 ⊢ ((𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin^II))
5	2, 4	syl 17	. . . 4 ⊢ ((𝐴 ∈ Fin^Ia ∧ 𝑏 ⊆ 𝐴) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin^II))
6	1, 5	sylan2 593	. . 3 ⊢ ((𝐴 ∈ Fin^Ia ∧ 𝑏 ∈ 𝒫 𝐴) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin^II))
7	6	ralrimiva 3124	. 2 ⊢ (𝐴 ∈ Fin^Ia → ∀𝑏 ∈ 𝒫 𝐴(𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin^II))
8		fin1a2s 10305	. 2 ⊢ ((𝐴 ∈ Fin^Ia ∧ ∀𝑏 ∈ 𝒫 𝐴(𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin^II)) → 𝐴 ∈ Fin^II)
9	7, 8	mpdan 687	1 ⊢ (𝐴 ∈ Fin^Ia → 𝐴 ∈ Fin^II)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 ∈ wcel 2111 ∀wral 3047 ∖ cdif 3899 ⊆ wss 3902 𝒫 cpw 4550 Fincfn 8869 Fin^Iacfin1a 10169 Fin^IIcfin2 10170

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-rpss 7656 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-seqom 8367 df-1o 8385 df-2o 8386 df-oadd 8389 df-omul 8390 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-wdom 9451 df-card 9832 df-fin1a 10176 df-fin2 10177 df-fin4 10178 df-fin3 10179

This theorem is referenced by: (None)

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