Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3748 | . 2 ⊢ ((V ∖ dom card) ⊆ Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin) | |
2 | dfac10 8842 | . . . 4 ⊢ (CHOICE ↔ dom card = V) | |
3 | finnum 8657 | . . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ∈ dom card) | |
4 | 3 | ssriv 3572 | . . . . . 6 ⊢ Fin ⊆ dom card |
5 | ssequn2 3748 | . . . . . 6 ⊢ (Fin ⊆ dom card ↔ (dom card ∪ Fin) = dom card) | |
6 | 4, 5 | mpbi 219 | . . . . 5 ⊢ (dom card ∪ Fin) = dom card |
7 | 6 | eqeq1i 2615 | . . . 4 ⊢ ((dom card ∪ Fin) = V ↔ dom card = V) |
8 | 2, 7 | bitr4i 266 | . . 3 ⊢ (CHOICE ↔ (dom card ∪ Fin) = V) |
9 | ssv 3588 | . . . 4 ⊢ (dom card ∪ Fin) ⊆ V | |
10 | eqss 3583 | . . . 4 ⊢ ((dom card ∪ Fin) = V ↔ ((dom card ∪ Fin) ⊆ V ∧ V ⊆ (dom card ∪ Fin))) | |
11 | 9, 10 | mpbiran 955 | . . 3 ⊢ ((dom card ∪ Fin) = V ↔ V ⊆ (dom card ∪ Fin)) |
12 | ssundif 4004 | . . 3 ⊢ (V ⊆ (dom card ∪ Fin) ↔ (V ∖ dom card) ⊆ Fin) | |
13 | 8, 11, 12 | 3bitri 285 | . 2 ⊢ (CHOICE ↔ (V ∖ dom card) ⊆ Fin) |
14 | dffin7-2 9103 | . . 3 ⊢ FinVII = (Fin ∪ (V ∖ dom card)) | |
15 | 14 | eqeq1i 2615 | . 2 ⊢ (FinVII = Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin) |
16 | 1, 13, 15 | 3bitr4i 291 | 1 ⊢ (CHOICE ↔ FinVII = Fin) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 dom cdm 5038 Fincfn 7841 cardccrd 8644 CHOICEwac 8821 FinVIIcfin7 8989 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-om 6958 df-wrecs 7294 df-recs 7355 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-ac 8822 df-fin7 8996 |
This theorem is referenced by: fin71ac 9236 |
Copyright terms: Public domain | W3C validator |