fin41 - Metamath Proof Explorer

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Theorem fin41 10532

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 3457	. . . . 5 ⊢ 𝑥 ∈ V
2	1	domtriom 10531	. . . 4 ⊢ (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω)
3	2	con2bii 360	. . 3 ⊢ (𝑥 ≺ ω ↔ ¬ ω ≼ 𝑥)
4		isfinite 9663	. . 3 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
5		isfin4-2 10402	. . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ Fin^IV ↔ ¬ ω ≼ 𝑥))
6	5	elv 3458	. . 3 ⊢ (𝑥 ∈ Fin^IV ↔ ¬ ω ≼ 𝑥)
7	3, 4, 6	3bitr4ri 307	. 2 ⊢ (𝑥 ∈ Fin^IV ↔ 𝑥 ∈ Fin)
8	7	eqriv 2760	1 ⊢ Fin^IV = Fin

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 ↔ wb 209 = wceq 1569 ∈ wcel 2146 Vcvv 3453 class class class wbr 5105 ωcom 7884 ≼ cdom 8982 ≺ csdm 8983 Fincfn 8984 Fin^IVcfin4 10368

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-inf2 9652 ax-cc 10523

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-op 4838 df-uni 4870 df-int 4910 df-iun 4955 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7430 df-oprab 7431 df-mpo 7432 df-om 7885 df-1st 8008 df-2nd 8009 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-1o 8487 df-2o 8488 df-oadd 8491 df-er 8728 df-map 8860 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9992 df-card 10030 df-fin4 10375

This theorem is used by: (None)

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