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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 3991 | . . . 4 ⊢ dom card ⊆ V | |
2 | eqss 3982 | . . . 4 ⊢ (dom card = V ↔ (dom card ⊆ V ∧ V ⊆ dom card)) | |
3 | 1, 2 | mpbiran 707 | . . 3 ⊢ (dom card = V ↔ V ⊆ dom card) |
4 | dfac10 9563 | . . 3 ⊢ (CHOICE ↔ dom card = V) | |
5 | unir1 9242 | . . . 4 ⊢ ∪ (𝑅1 “ On) = V | |
6 | 5 | sseq1i 3995 | . . 3 ⊢ (∪ (𝑅1 “ On) ⊆ dom card ↔ V ⊆ dom card) |
7 | 3, 4, 6 | 3bitr4i 305 | . 2 ⊢ (CHOICE ↔ ∪ (𝑅1 “ On) ⊆ dom card) |
8 | dfac12r 9572 | . 2 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ ∪ (𝑅1 “ On) ⊆ dom card) | |
9 | 7, 8 | bitr4i 280 | 1 ⊢ (CHOICE ↔ ∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4838 dom cdm 5555 “ cima 5558 Oncon0 6191 𝑅1cr1 9191 cardccrd 9364 CHOICEwac 9541 |
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-reg 9056 ax-inf2 9104 |
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-se 5515 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-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-oadd 8106 df-omul 8107 df-er 8289 df-en 8510 df-dom 8511 df-oi 8974 df-har 9022 df-r1 9193 df-rank 9194 df-card 9368 df-ac 9542 |
This theorem is referenced by: dfac12 9575 |
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