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Mirrors > Home > MPE Home > Th. List > dfac12a | Structured version Visualization version GIF version |
Description: The axiom of choice holds iff every ordinal has a well-orderable powerset. (Contributed by Mario Carneiro, 29-May-2015.) |
Ref | Expression |
---|---|
dfac12a | ⊢ (CHOICE ↔ ∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3939 | . . . 4 ⊢ dom card ⊆ V | |
2 | eqss 3930 | . . . 4 ⊢ (dom card = V ↔ (dom card ⊆ V ∧ V ⊆ dom card)) | |
3 | 1, 2 | mpbiran 708 | . . 3 ⊢ (dom card = V ↔ V ⊆ dom card) |
4 | dfac10 9548 | . . 3 ⊢ (CHOICE ↔ dom card = V) | |
5 | unir1 9226 | . . . 4 ⊢ ∪ (𝑅1 “ On) = V | |
6 | 5 | sseq1i 3943 | . . 3 ⊢ (∪ (𝑅1 “ On) ⊆ dom card ↔ V ⊆ dom card) |
7 | 3, 4, 6 | 3bitr4i 306 | . 2 ⊢ (CHOICE ↔ ∪ (𝑅1 “ On) ⊆ dom card) |
8 | dfac12r 9557 | . 2 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ ∪ (𝑅1 “ On) ⊆ dom card) | |
9 | 7, 8 | bitr4i 281 | 1 ⊢ (CHOICE ↔ ∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ⊆ wss 3881 𝒫 cpw 4497 ∪ cuni 4800 dom cdm 5519 “ cima 5522 Oncon0 6159 𝑅1cr1 9175 cardccrd 9348 CHOICEwac 9526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-reg 9040 ax-inf2 9088 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-oadd 8089 df-omul 8090 df-er 8272 df-en 8493 df-dom 8494 df-oi 8958 df-har 9005 df-r1 9177 df-rank 9178 df-card 9352 df-ac 9527 |
This theorem is referenced by: dfac12 9560 |
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