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Mirrors > Home > MPE Home > Th. List > acacni | Structured version Visualization version GIF version |
Description: A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
acacni | ⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → AC 𝐴 = V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . . . 4 ⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
2 | vex 3445 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | simpl 483 | . . . . . 6 ⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → CHOICE) | |
4 | dfac10 9994 | . . . . . 6 ⊢ (CHOICE ↔ dom card = V) | |
5 | 3, 4 | sylib 217 | . . . . 5 ⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → dom card = V) |
6 | 2, 5 | eleqtrrid 2844 | . . . 4 ⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → 𝑥 ∈ dom card) |
7 | numacn 9906 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ dom card → 𝑥 ∈ AC 𝐴)) | |
8 | 1, 6, 7 | sylc 65 | . . 3 ⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → 𝑥 ∈ AC 𝐴) |
9 | 2 | a1i 11 | . . 3 ⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → 𝑥 ∈ V) |
10 | 8, 9 | 2thd 264 | . 2 ⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ AC 𝐴 ↔ 𝑥 ∈ V)) |
11 | 10 | eqrdv 2734 | 1 ⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → AC 𝐴 = V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 Vcvv 3441 dom cdm 5620 cardccrd 9792 AC wacn 9795 CHOICEwac 9972 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-se 5576 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-isom 6488 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-er 8569 df-map 8688 df-en 8805 df-dom 8806 df-card 9796 df-acn 9799 df-ac 9973 |
This theorem is referenced by: dfacacn 9998 dfac13 9999 ptcls 22873 dfac14 22875 |
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