fin41 - Metamath Proof Explorer

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Theorem fin41 10401

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 3458	. . . . 5 ⊢ 𝑥 ∈ V
2	1	domtriom 10400	. . . 4 ⊢ (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω)
3	2	con2bii 359	. . 3 ⊢ (𝑥 ≺ ω ↔ ¬ ω ≼ 𝑥)
4		isfinite 9607	. . 3 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
5		isfin4-2 10271	. . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ Fin^IV ↔ ¬ ω ≼ 𝑥))
6	5	elv 3459	. . 3 ⊢ (𝑥 ∈ Fin^IV ↔ ¬ ω ≼ 𝑥)
7	3, 4, 6	3bitr4ri 306	. 2 ⊢ (𝑥 ∈ Fin^IV ↔ 𝑥 ∈ Fin)
8	7	eqriv 2759	1 ⊢ Fin^IV = Fin

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1560 ∈ wcel 2142 Vcvv 3454 class class class wbr 5100 ωcom 7846 ≼ cdom 8925 ≺ csdm 8926 Fincfn 8927 Fin^IVcfin4 10237

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-inf2 9596 ax-cc 10392

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-er 8678 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-dju 9859 df-card 9897 df-fin4 10244

This theorem is referenced by: (None)

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