fin1a2 - Metamath Proof Explorer

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Theorem fin1a2 10337

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 4548	. . . 4 ⊢ (𝑏 ∈ 𝒫 𝐴 → 𝑏 ⊆ 𝐴)
2		fin1ai 10215	. . . . 5 ⊢ ((𝐴 ∈ Fin^Ia ∧ 𝑏 ⊆ 𝐴) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin))
3		fin12 10335	. . . . . 6 ⊢ ((𝐴 ∖ 𝑏) ∈ Fin → (𝐴 ∖ 𝑏) ∈ Fin^II)
4	3	orim2i 911	. . . . 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 594	. . 3 ⊢ ((𝐴 ∈ Fin^Ia ∧ 𝑏 ∈ 𝒫 𝐴) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin^II))
7	6	ralrimiva 3129	. 2 ⊢ (𝐴 ∈ Fin^Ia → ∀𝑏 ∈ 𝒫 𝐴(𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin^II))
8		fin1a2s 10336	. 2 ⊢ ((𝐴 ∈ Fin^Ia ∧ ∀𝑏 ∈ 𝒫 𝐴(𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin^II)) → 𝐴 ∈ Fin^II)
9	7, 8	mpdan 688	1 ⊢ (𝐴 ∈ Fin^Ia → 𝐴 ∈ Fin^II)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 ∈ wcel 2114 ∀wral 3051 ∖ cdif 3886 ⊆ wss 3889 𝒫 cpw 4541 Fincfn 8893 Fin^Iacfin1a 10200 Fin^IIcfin2 10201

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-rpss 7677 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-seqom 8387 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-wdom 9480 df-card 9863 df-fin1a 10207 df-fin2 10208 df-fin4 10209 df-fin3 10210

This theorem is referenced by: (None)

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