![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dfac13 | Structured version Visualization version Unicode version |
Description: The axiom of choice holds iff every set has choice sequences as long as itself. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
dfac13 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3050 |
. . . 4
![]() ![]() ![]() ![]() | |
2 | acacni 8575 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | 1, 2 | mpan2 678 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | 1, 3 | syl5eleqr 2538 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | 4 | alrimiv 1775 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | vex 3050 |
. . . . . . . . 9
![]() ![]() ![]() ![]() | |
7 | 6 | pwex 4589 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() |
8 | id 22 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | acneq 8479 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | 8, 9 | eleq12d 2525 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | 7, 10 | spcv 3142 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | vex 3050 |
. . . . . . . 8
![]() ![]() ![]() ![]() | |
13 | 6 | canth2 7730 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() |
14 | sdomdom 7602 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | acndom2 8490 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 13, 14, 15 | mp2b 10 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | acnnum 8488 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | 16, 17 | sylib 200 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | numacn 8485 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 12, 18, 19 | mpsyl 65 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 11, 20 | syl 17 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 6 | a1i 11 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 21, 22 | 2thd 244 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 23 | eqrdv 2451 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | alrimiv 1775 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | dfacacn 8576 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 25, 26 | sylibr 216 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 5, 27 | impbii 191 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-rep 4518 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-reu 2746 df-rmo 2747 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-pss 3422 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-tp 3975 df-op 3977 df-uni 4202 df-int 4238 df-iun 4283 df-br 4406 df-opab 4465 df-mpt 4466 df-tr 4501 df-eprel 4748 df-id 4752 df-po 4758 df-so 4759 df-fr 4796 df-se 4797 df-we 4798 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-pred 5383 df-ord 5429 df-on 5430 df-lim 5431 df-suc 5432 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-isom 5594 df-riota 6257 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-om 6698 df-1st 6798 df-2nd 6799 df-wrecs 7033 df-recs 7095 df-1o 7187 df-er 7368 df-map 7479 df-en 7575 df-dom 7576 df-sdom 7577 df-fin 7578 df-card 8378 df-acn 8381 df-ac 8552 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |