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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 3993 | . . . 4 ⊢ dom card ⊆ V | |
2 | eqss 3984 | . . . 4 ⊢ (dom card = V ↔ (dom card ⊆ V ∧ V ⊆ dom card)) | |
3 | 1, 2 | mpbiran 707 | . . 3 ⊢ (dom card = V ↔ V ⊆ dom card) |
4 | dfac10 9565 | . . 3 ⊢ (CHOICE ↔ dom card = V) | |
5 | unir1 9244 | . . . 4 ⊢ ∪ (𝑅1 “ On) = V | |
6 | 5 | sseq1i 3997 | . . 3 ⊢ (∪ (𝑅1 “ On) ⊆ dom card ↔ V ⊆ dom card) |
7 | 3, 4, 6 | 3bitr4i 305 | . 2 ⊢ (CHOICE ↔ ∪ (𝑅1 “ On) ⊆ dom card) |
8 | dfac12r 9574 | . 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 3140 Vcvv 3496 ⊆ wss 3938 𝒫 cpw 4541 ∪ cuni 4840 dom cdm 5557 “ cima 5560 Oncon0 6193 𝑅1cr1 9193 cardccrd 9366 CHOICEwac 9543 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-reg 9058 ax-inf2 9106 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-oadd 8108 df-omul 8109 df-er 8291 df-en 8512 df-dom 8513 df-oi 8976 df-har 9024 df-r1 9195 df-rank 9196 df-card 9370 df-ac 9544 |
This theorem is referenced by: dfac12 9577 |
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