fin1a2 - Metamath Proof Explorer

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Theorem fin1a2 10375

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 4573	. . . 4 ⊢ (𝑏 ∈ 𝒫 𝐴 → 𝑏 ⊆ 𝐴)
2		fin1ai 10253	. . . . 5 ⊢ ((𝐴 ∈ Fin^Ia ∧ 𝑏 ⊆ 𝐴) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin))
3		fin12 10373	. . . . . 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 3126	. 2 ⊢ (𝐴 ∈ Fin^Ia → ∀𝑏 ∈ 𝒫 𝐴(𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin^II))
8		fin1a2s 10374	. 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 2109 ∀wral 3045 ∖ cdif 3914 ⊆ wss 3917 𝒫 cpw 4566 Fincfn 8921 Fin^Iacfin1a 10238 Fin^IIcfin2 10239

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-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 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-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 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-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 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-pred 6277 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-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-rpss 7702 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-seqom 8419 df-1o 8437 df-2o 8438 df-oadd 8441 df-omul 8442 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-wdom 9525 df-card 9899 df-fin1a 10245 df-fin2 10246 df-fin4 10247 df-fin3 10248

This theorem is referenced by: (None)

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