Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 476 | . . . 4 ⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
2 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | simpl 472 | . . . . . 6 ⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → CHOICE) | |
4 | dfac10 8842 | . . . . . 6 ⊢ (CHOICE ↔ dom card = V) | |
5 | 3, 4 | sylib 207 | . . . . 5 ⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → dom card = V) |
6 | 2, 5 | syl5eleqr 2695 | . . . 4 ⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → 𝑥 ∈ dom card) |
7 | numacn 8755 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ dom card → 𝑥 ∈ AC 𝐴)) | |
8 | 1, 6, 7 | sylc 63 | . . 3 ⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → 𝑥 ∈ AC 𝐴) |
9 | 2 | a1i 11 | . . 3 ⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → 𝑥 ∈ V) |
10 | 8, 9 | 2thd 254 | . 2 ⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ AC 𝐴 ↔ 𝑥 ∈ V)) |
11 | 10 | eqrdv 2608 | 1 ⊢ ((CHOICE ∧ 𝐴 ∈ 𝑉) → AC 𝐴 = V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 dom cdm 5038 cardccrd 8644 AC wacn 8647 CHOICEwac 8821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-card 8648 df-acn 8651 df-ac 8822 |
This theorem is referenced by: dfacacn 8846 dfac13 8847 ptcls 21229 dfac14 21231 |
Copyright terms: Public domain | W3C validator |