fin41 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > fin41		Structured version Visualization version GIF version

Theorem fin41 9664

Description: Under countable choice, the IV-finite sets (Dedekind-finite) coincide with I-finite (finite in the usual sense) sets. (Contributed by Mario Carneiro, 16-May-2015.)

**Assertion**
Ref	Expression
fin41	⊢ Fin^IV = Fin

**Proof of Theorem fin41**
Step	Hyp	Ref	Expression
1		vex 3419	. . . . 5 ⊢ 𝑥 ∈ V
2	1	domtriom 9663	. . . 4 ⊢ (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω)
3	2	con2bii 350	. . 3 ⊢ (𝑥 ≺ ω ↔ ¬ ω ≼ 𝑥)
4		isfinite 8909	. . 3 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
5		isfin4-2 9534	. . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ Fin^IV ↔ ¬ ω ≼ 𝑥))
6	5	elv 3421	. . 3 ⊢ (𝑥 ∈ Fin^IV ↔ ¬ ω ≼ 𝑥)
7	3, 4, 6	3bitr4ri 296	. 2 ⊢ (𝑥 ∈ Fin^IV ↔ 𝑥 ∈ Fin)
8	7	eqriv 2776	1 ⊢ Fin^IV = Fin

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 198 = wceq 1507 ∈ wcel 2050 Vcvv 3416 class class class wbr 4929 ωcom 7396 ≼ cdom 8304 ≺ csdm 8305 Fincfn 8306 Fin^IVcfin4 9500

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 ax-cc 9655

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-2o 7906 df-oadd 7909 df-er 8089 df-map 8208 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-dju 9124 df-card 9162 df-fin4 9507

This theorem is referenced by: (None)

W3C validator