fin1a2 - Metamath Proof Explorer

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Theorem fin1a2 10417

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 4574	. . . 4 ⊢ (𝑏 ∈ 𝒫 𝐴 → 𝑏 ⊆ 𝐴)
2		fin1ai 10295	. . . . 5 ⊢ ((𝐴 ∈ Fin^Ia ∧ 𝑏 ⊆ 𝐴) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin))
3		fin12 10415	. . . . . 6 ⊢ ((𝐴 ∖ 𝑏) ∈ Fin → (𝐴 ∖ 𝑏) ∈ Fin^II)
4	3	orim2i 924	. . . . 5 ⊢ ((𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin^II))
5	2, 4	syl 18	. . . 4 ⊢ ((𝐴 ∈ Fin^Ia ∧ 𝑏 ⊆ 𝐴) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin^II))
6	1, 5	sylan2 605	. . 3 ⊢ ((𝐴 ∈ Fin^Ia ∧ 𝑏 ∈ 𝒫 𝐴) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin^II))
7	6	ralrimiva 3160	. 2 ⊢ (𝐴 ∈ Fin^Ia → ∀𝑏 ∈ 𝒫 𝐴(𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin^II))
8		fin1a2s 10416	. 2 ⊢ ((𝐴 ∈ Fin^Ia ∧ ∀𝑏 ∈ 𝒫 𝐴(𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin^II)) → 𝐴 ∈ Fin^II)
9	7, 8	mpdan 700	1 ⊢ (𝐴 ∈ Fin^Ia → 𝐴 ∈ Fin^II)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ wo 861 ∈ wcel 2146 ∀wral 3082 ∖ cdif 3905 ⊆ wss 3908 𝒫 cpw 4567 Fincfn 8952 Fin^Iacfin1a 10280 Fin^IIcfin2 10281

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-rpss 7733 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-seqom 8444 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8703 df-map 8835 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-wdom 9537 df-card 9944 df-fin1a 10287 df-fin2 10288 df-fin4 10289 df-fin3 10290

This theorem is used by: (None)

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