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Mirrors > Home > MPE Home > Th. List > fin41 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
fin41 | ⊢ FinIV = Fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3450 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | domtriom 10380 | . . . 4 ⊢ (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω) |
3 | 2 | con2bii 358 | . . 3 ⊢ (𝑥 ≺ ω ↔ ¬ ω ≼ 𝑥) |
4 | isfinite 9589 | . . 3 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω) | |
5 | isfin4-2 10251 | . . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ FinIV ↔ ¬ ω ≼ 𝑥)) | |
6 | 5 | elv 3452 | . . 3 ⊢ (𝑥 ∈ FinIV ↔ ¬ ω ≼ 𝑥) |
7 | 3, 4, 6 | 3bitr4ri 304 | . 2 ⊢ (𝑥 ∈ FinIV ↔ 𝑥 ∈ Fin) |
8 | 7 | eqriv 2734 | 1 ⊢ FinIV = Fin |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1542 ∈ wcel 2107 Vcvv 3446 class class class wbr 5106 ωcom 7803 ≼ cdom 8882 ≺ csdm 8883 Fincfn 8884 FinIVcfin4 10217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9578 ax-cc 10372 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-oadd 8417 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-dju 9838 df-card 9876 df-fin4 10224 |
This theorem is referenced by: (None) |
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