fin41 - Metamath Proof Explorer

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Theorem fin41 9866

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 3497	. . . . 5 ⊢ 𝑥 ∈ V
2	1	domtriom 9865	. . . 4 ⊢ (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω)
3	2	con2bii 360	. . 3 ⊢ (𝑥 ≺ ω ↔ ¬ ω ≼ 𝑥)
4		isfinite 9115	. . 3 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
5		isfin4-2 9736	. . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ Fin^IV ↔ ¬ ω ≼ 𝑥))
6	5	elv 3499	. . 3 ⊢ (𝑥 ∈ Fin^IV ↔ ¬ ω ≼ 𝑥)
7	3, 4, 6	3bitr4ri 306	. 2 ⊢ (𝑥 ∈ Fin^IV ↔ 𝑥 ∈ Fin)
8	7	eqriv 2818	1 ⊢ Fin^IV = Fin

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1537 ∈ wcel 2114 Vcvv 3494 class class class wbr 5066 ωcom 7580 ≼ cdom 8507 ≺ csdm 8508 Fincfn 8509 Fin^IVcfin4 9702

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cc 9857

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 df-card 9368 df-fin4 9709

This theorem is referenced by: (None)

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