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 9149

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 3176	. . . . 5 ⊢ 𝑥 ∈ V
2	1	domtriom 9148	. . . 4 ⊢ (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω)
3	2	con2bii 346	. . 3 ⊢ (𝑥 ≺ ω ↔ ¬ ω ≼ 𝑥)
4		isfinite 8432	. . 3 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
5		isfin4-2 9019	. . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ Fin^IV ↔ ¬ ω ≼ 𝑥))
6	1, 5	ax-mp 5	. . 3 ⊢ (𝑥 ∈ Fin^IV ↔ ¬ ω ≼ 𝑥)
7	3, 4, 6	3bitr4ri 292	. 2 ⊢ (𝑥 ∈ Fin^IV ↔ 𝑥 ∈ Fin)
8	7	eqriv 2607	1 ⊢ Fin^IV = Fin

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 ωcom 6957 ≼ cdom 7839 ≺ csdm 7840 Fincfn 7841 Fin^IVcfin4 8985

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-fin4 8992

This theorem is referenced by: (None)

W3C validator