fin1a2 - Metamath Proof Explorer

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Theorem fin1a2 9439

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 4307	. . . 4 ⊢ (𝑏 ∈ 𝒫 𝐴 → 𝑏 ⊆ 𝐴)
2		fin1ai 9317	. . . . 5 ⊢ ((𝐴 ∈ Fin^Ia ∧ 𝑏 ⊆ 𝐴) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin))
3		fin12 9437	. . . . . 6 ⊢ ((𝐴 ∖ 𝑏) ∈ Fin → (𝐴 ∖ 𝑏) ∈ Fin^II)
4	3	orim2i 894	. . . . 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 580	. . 3 ⊢ ((𝐴 ∈ Fin^Ia ∧ 𝑏 ∈ 𝒫 𝐴) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin^II))
7	6	ralrimiva 3115	. 2 ⊢ (𝐴 ∈ Fin^Ia → ∀𝑏 ∈ 𝒫 𝐴(𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin^II))
8		fin1a2s 9438	. 2 ⊢ ((𝐴 ∈ Fin^Ia ∧ ∀𝑏 ∈ 𝒫 𝐴(𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin^II)) → 𝐴 ∈ Fin^II)
9	7, 8	mpdan 667	1 ⊢ (𝐴 ∈ Fin^Ia → 𝐴 ∈ Fin^II)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 382 ∨ wo 834 ∈ wcel 2145 ∀wral 3061 ∖ cdif 3720 ⊆ wss 3723 𝒫 cpw 4297 Fincfn 8109 Fin^Iacfin1a 9302 Fin^IIcfin2 9303

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096

This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-rpss 7084 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-seqom 7696 df-1o 7713 df-2o 7714 df-oadd 7717 df-omul 7718 df-er 7896 df-map 8011 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-wdom 8620 df-card 8965 df-fin1a 9309 df-fin2 9310 df-fin4 9311 df-fin3 9312

This theorem is referenced by: (None)

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