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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 3851 | . . . 4 ⊢ dom card ⊆ V | |
2 | eqss 3843 | . . . 4 ⊢ (dom card = V ↔ (dom card ⊆ V ∧ V ⊆ dom card)) | |
3 | 1, 2 | mpbiran 702 | . . 3 ⊢ (dom card = V ↔ V ⊆ dom card) |
4 | dfac10 9275 | . . 3 ⊢ (CHOICE ↔ dom card = V) | |
5 | unir1 8954 | . . . 4 ⊢ ∪ (𝑅1 “ On) = V | |
6 | 5 | sseq1i 3855 | . . 3 ⊢ (∪ (𝑅1 “ On) ⊆ dom card ↔ V ⊆ dom card) |
7 | 3, 4, 6 | 3bitr4i 295 | . 2 ⊢ (CHOICE ↔ ∪ (𝑅1 “ On) ⊆ dom card) |
8 | dfac12r 9284 | . 2 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ ∪ (𝑅1 “ On) ⊆ dom card) | |
9 | 7, 8 | bitr4i 270 | 1 ⊢ (CHOICE ↔ ∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1658 ∈ wcel 2166 ∀wral 3118 Vcvv 3415 ⊆ wss 3799 𝒫 cpw 4379 ∪ cuni 4659 dom cdm 5343 “ cima 5346 Oncon0 5964 𝑅1cr1 8903 cardccrd 9075 CHOICEwac 9252 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-reg 8767 ax-inf2 8816 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-se 5303 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-isom 6133 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-oadd 7831 df-omul 7832 df-er 8010 df-en 8224 df-dom 8225 df-oi 8685 df-har 8733 df-r1 8905 df-rank 8906 df-card 9079 df-ac 9253 |
This theorem is referenced by: dfac12 9287 |
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