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Mirrors > Home > MPE Home > Th. List > dfacfin7 | Structured version Visualization version GIF version |
Description: Axiom of Choice equivalent: the VII-finite sets are the same as I-finite sets. (Contributed by Mario Carneiro, 18-May-2015.) |
Ref | Expression |
---|---|
dfacfin7 | ⊢ (CHOICE ↔ FinVII = Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssequn2 4156 | . 2 ⊢ ((V ∖ dom card) ⊆ Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin) | |
2 | dfac10 9551 | . . . 4 ⊢ (CHOICE ↔ dom card = V) | |
3 | finnum 9365 | . . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ∈ dom card) | |
4 | 3 | ssriv 3968 | . . . . . 6 ⊢ Fin ⊆ dom card |
5 | ssequn2 4156 | . . . . . 6 ⊢ (Fin ⊆ dom card ↔ (dom card ∪ Fin) = dom card) | |
6 | 4, 5 | mpbi 231 | . . . . 5 ⊢ (dom card ∪ Fin) = dom card |
7 | 6 | eqeq1i 2823 | . . . 4 ⊢ ((dom card ∪ Fin) = V ↔ dom card = V) |
8 | 2, 7 | bitr4i 279 | . . 3 ⊢ (CHOICE ↔ (dom card ∪ Fin) = V) |
9 | ssv 3988 | . . . 4 ⊢ (dom card ∪ Fin) ⊆ V | |
10 | eqss 3979 | . . . 4 ⊢ ((dom card ∪ Fin) = V ↔ ((dom card ∪ Fin) ⊆ V ∧ V ⊆ (dom card ∪ Fin))) | |
11 | 9, 10 | mpbiran 705 | . . 3 ⊢ ((dom card ∪ Fin) = V ↔ V ⊆ (dom card ∪ Fin)) |
12 | ssundif 4429 | . . 3 ⊢ (V ⊆ (dom card ∪ Fin) ↔ (V ∖ dom card) ⊆ Fin) | |
13 | 8, 11, 12 | 3bitri 298 | . 2 ⊢ (CHOICE ↔ (V ∖ dom card) ⊆ Fin) |
14 | dffin7-2 9808 | . . 3 ⊢ FinVII = (Fin ∪ (V ∖ dom card)) | |
15 | 14 | eqeq1i 2823 | . 2 ⊢ (FinVII = Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin) |
16 | 1, 13, 15 | 3bitr4i 304 | 1 ⊢ (CHOICE ↔ FinVII = Fin) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 Vcvv 3492 ∖ cdif 3930 ∪ cun 3931 ⊆ wss 3933 dom cdm 5548 Fincfn 8497 cardccrd 9352 CHOICEwac 9529 FinVIIcfin7 9694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-om 7570 df-wrecs 7936 df-recs 7997 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-ac 9530 df-fin7 9701 |
This theorem is referenced by: fin71ac 9943 |
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