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Mirrors > Home > ILE Home > Th. List > 4bc2eq6 | Unicode version |
Description: The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.) |
Ref | Expression |
---|---|
4bc2eq6 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 8824 |
. . . . 5
![]() ![]() ![]() ![]() | |
2 | 4z 8843 |
. . . . 5
![]() ![]() ![]() ![]() | |
3 | 2z 8841 |
. . . . 5
![]() ![]() ![]() ![]() | |
4 | 1, 2, 3 | 3pm3.2i 1122 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | 0le2 8575 |
. . . . 5
![]() ![]() ![]() ![]() | |
6 | 2re 8555 |
. . . . . 6
![]() ![]() ![]() ![]() | |
7 | 4re 8562 |
. . . . . 6
![]() ![]() ![]() ![]() | |
8 | 2lt4 8652 |
. . . . . 6
![]() ![]() ![]() ![]() | |
9 | 6, 7, 8 | ltleii 7650 |
. . . . 5
![]() ![]() ![]() ![]() |
10 | 5, 9 | pm3.2i 267 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | elfz4 9496 |
. . . 4
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12 | 4, 10, 11 | mp2an 418 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | bcval2 10221 |
. . 3
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14 | 12, 13 | ax-mp 7 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 3nn0 8754 |
. . . . . 6
![]() ![]() ![]() ![]() | |
16 | facp1 10201 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 15, 16 | ax-mp 7 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | df-4 8546 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 18 | fveq2i 5323 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 18 | oveq2i 5679 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 17, 19, 20 | 3eqtr4i 2119 |
. . . 4
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22 | 4cn 8563 |
. . . . . . . . 9
![]() ![]() ![]() ![]() | |
23 | 2cn 8556 |
. . . . . . . . 9
![]() ![]() ![]() ![]() | |
24 | 2p2e4 8606 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 22, 23, 23, 24 | subaddrii 7834 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 25 | fveq2i 5323 |
. . . . . . 7
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27 | fac2 10202 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 26, 27 | eqtri 2109 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 28, 27 | oveq12i 5680 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 2t2e4 8633 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | 29, 30 | eqtri 2109 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 21, 31 | oveq12i 5680 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | faccl 10206 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
34 | 15, 33 | ax-mp 7 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | 34 | nncni 8495 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | 4ap0 8584 |
. . . . 5
![]() ![]() ![]() | |
37 | 35, 22, 36 | divcanap4i 8289 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | fac3 10203 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
39 | 37, 38 | eqtri 2109 |
. . 3
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40 | 32, 39 | eqtri 2109 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | 14, 40 | eqtri 2109 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3962 ax-sep 3965 ax-nul 3973 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-iinf 4418 ax-cnex 7499 ax-resscn 7500 ax-1cn 7501 ax-1re 7502 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-mulrcl 7507 ax-addcom 7508 ax-mulcom 7509 ax-addass 7510 ax-mulass 7511 ax-distr 7512 ax-i2m1 7513 ax-0lt1 7514 ax-1rid 7515 ax-0id 7516 ax-rnegex 7517 ax-precex 7518 ax-cnre 7519 ax-pre-ltirr 7520 ax-pre-ltwlin 7521 ax-pre-lttrn 7522 ax-pre-apti 7523 ax-pre-ltadd 7524 ax-pre-mulgt0 7525 ax-pre-mulext 7526 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2624 df-sbc 2844 df-csb 2937 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-nul 3290 df-if 3400 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-int 3697 df-iun 3740 df-br 3854 df-opab 3908 df-mpt 3909 df-tr 3945 df-id 4131 df-po 4134 df-iso 4135 df-iord 4204 df-on 4206 df-ilim 4207 df-suc 4209 df-iom 4421 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-iota 4995 df-fun 5032 df-fn 5033 df-f 5034 df-f1 5035 df-fo 5036 df-f1o 5037 df-fv 5038 df-riota 5624 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-1st 5927 df-2nd 5928 df-recs 6086 df-frec 6172 df-pnf 7587 df-mnf 7588 df-xr 7589 df-ltxr 7590 df-le 7591 df-sub 7718 df-neg 7719 df-reap 8115 df-ap 8122 df-div 8203 df-inn 8486 df-2 8544 df-3 8545 df-4 8546 df-5 8547 df-6 8548 df-n0 8737 df-z 8814 df-uz 9083 df-q 9168 df-fz 9488 df-iseq 9916 df-fac 10197 df-bc 10219 |
This theorem is referenced by: ex-bc 11960 |
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