![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fin1a2 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
fin1a2 | ⊢ (𝐴 ∈ FinIa → 𝐴 ∈ FinII) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwi 4567 | . . . 4 ⊢ (𝑏 ∈ 𝒫 𝐴 → 𝑏 ⊆ 𝐴) | |
2 | fin1ai 10228 | . . . . 5 ⊢ ((𝐴 ∈ FinIa ∧ 𝑏 ⊆ 𝐴) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin)) | |
3 | fin12 10348 | . . . . . 6 ⊢ ((𝐴 ∖ 𝑏) ∈ Fin → (𝐴 ∖ 𝑏) ∈ FinII) | |
4 | 3 | orim2i 909 | . . . . 5 ⊢ ((𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ FinII)) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ FinIa ∧ 𝑏 ⊆ 𝐴) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ FinII)) |
6 | 1, 5 | sylan2 593 | . . 3 ⊢ ((𝐴 ∈ FinIa ∧ 𝑏 ∈ 𝒫 𝐴) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ FinII)) |
7 | 6 | ralrimiva 3143 | . 2 ⊢ (𝐴 ∈ FinIa → ∀𝑏 ∈ 𝒫 𝐴(𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ FinII)) |
8 | fin1a2s 10349 | . 2 ⊢ ((𝐴 ∈ FinIa ∧ ∀𝑏 ∈ 𝒫 𝐴(𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ FinII)) → 𝐴 ∈ FinII) | |
9 | 7, 8 | mpdan 685 | 1 ⊢ (𝐴 ∈ FinIa → 𝐴 ∈ FinII) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 845 ∈ wcel 2106 ∀wral 3064 ∖ cdif 3907 ⊆ wss 3910 𝒫 cpw 4560 Fincfn 8882 FinIacfin1a 10213 FinIIcfin2 10214 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-rpss 7659 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-seqom 8393 df-1o 8411 df-2o 8412 df-oadd 8415 df-omul 8416 df-er 8647 df-map 8766 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-wdom 9500 df-card 9874 df-fin1a 10220 df-fin2 10221 df-fin4 10222 df-fin3 10223 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |