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| Mirrors > Home > NFE Home > Th. List > cenc | Unicode version | ||
| Description: Cardinal exponentiation in terms of cardinality. Theorem XI.2.39 of [Rosser] p. 382. (Contributed by SF, 6-Mar-2015.) |
| Ref | Expression |
|---|---|
| cenc.1 |
    |
| cenc.2 |
 |
| Ref | Expression |
|---|---|
| cenc |
 Nc  1 ↑c Nc 1   Nc
 |
are mutually distinct and
distinct from all other variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elnc 6125 |
. . . . . . . . 9
1 Nc 1  1  1 | |
| 2 | enpw1 6062 |
. . . . . . . . 9
1 1 | |
| 3 | 1, 2 | bitr4i 243 |
. . . . . . . 8
1 Nc 1 |
| 4 | elnc 6125 |
. . . . . . . . 9
1
Nc 1 1
1 | |
| 5 | enpw1 6062 |
. . . . . . . . 9
1
1 | |
| 6 | 4, 5 | bitr4i 243 |
. . . . . . . 8
1
Nc 1 |
| 7 | enmap1 6074 |
. . . . . . . . 9
 | |
| 8 | enmap2 6068 |
. . . . . . . . 9
| |
| 9 | entr 6038 |
. . . . . . . . 9
![](wedge.gif "/\"){kind=small}
| |
| 10 | 7, 8, 9 | syl2an 463 |
. . . . . . . 8
|
| 11 | 3, 6, 10 | syl2anb 465 |
. . . . . . 7
1 Nc 1 1 Nc 1 |
| 12 | entr 6038 |
. . . . . . . 8
| |
| 13 | 12 | ancoms 439 |
. . . . . . 7
|
| 14 | 11, 13 | sylan 457 |
. . . . . 6
1 Nc 1 1 Nc 1
|
| 15 | 14 | 3impa 1146 |
. . . . 5
1 Nc 1 1 Nc 1 |
| 16 | 15 | exlimivv 1635 |
. . . 4
 1 Nc 1 1
Nc 1
|
| 17 | cenc.1 |
. . . . . . 7
| |
| 18 | 17 | pw1ex 4303 |
. . . . . 6
1 |
| 19 | 18 | ncelncsi 6121 |
. . . . 5
Nc 1 NC |
| 20 | cenc.2 |
. . . . . . 7
| |
| 21 | 20 | pw1ex 4303 |
. . . . . 6
1 |
| 22 | 21 | ncelncsi 6121 |
. . . . 5
Nc 1 NC |
| 23 | elce 6175 |
. . . . 5
Nc 1 NC Nc 1 NC Nc 1 ↑c Nc 1 1 Nc 1 1 Nc 1 | |
| 24 | 19, 22, 23 | mp2an 653 |
. . . 4
Nc 1 ↑c Nc 1 1 Nc 1 1 Nc 1 |
| 25 | elnc 6125 |
. . . 4
Nc
| |
| 26 | 16, 24, 25 | 3imtr4i 257 |
. . 3
Nc 1 ↑c Nc 1 Nc |
| 27 | 26 | ssriv 3277 |
. 2
Nc 1 ↑c Nc 1  Nc |
| 28 | 18 | ncid 6123 |
. . . . 5
1 Nc 1 |
| 29 | 21 | ncid 6123 |
. . . . 5
1 Nc 1 |
| 30 | pw1eq 4143 |
. . . . . . . . 9
1 1 | |
| 31 | 30 | eleq1d 2419 |
. . . . . . . 8
1 Nc 1 1 Nc 1 |
| 32 | 31 | adantr 451 |
. . . . . . 7
1
Nc 1 1 Nc 1 |
| 33 | pw1eq 4143 |
. . . . . . . . 9
1 1 | |
| 34 | 33 | eleq1d 2419 |
. . . . . . . 8
1 Nc 1 1 Nc 1 |
| 35 | 34 | adantl 452 |
. . . . . . 7
1
Nc 1 1 Nc 1 |
| 36 | oveq12 5532 |
. . . . . . . 8
| |
| 37 | 36 | breq2d 4651 |
. . . . . . 7
|
| 38 | 32, 35, 37 | 3anbi123d 1252 |
. . . . . 6
1 Nc 1 1 Nc 1
1 Nc 1 1 Nc 1
|
| 39 | 17, 20, 38 | spc2ev 2947 |
. . . . 5
1 Nc 1 1 Nc 1 1
Nc 1 1
Nc 1
|
| 40 | 28, 29, 39 | mp3an12 1267 |
. . . 4
1
Nc 1 1 Nc 1 |
| 41 | elce 6175 |
. . . . 5
Nc 1 NC Nc 1 NC Nc 1 ↑c Nc 1 1
Nc 1 1 Nc 1 | |
| 42 | 19, 22, 41 | mp2an 653 |
. . . 4
Nc 1 ↑c Nc 1 1
Nc 1 1 Nc 1 |
| 43 | 40, 25, 42 | 3imtr4i 257 |
. . 3
Nc
Nc 1 ↑c Nc 1 |
| 44 | 43 | ssriv 3277 |
. 2
Nc Nc 1 ↑c Nc 1 |
| 45 | 27, 44 | eqssi 3288 |
1
Nc 1 ↑c Nc 1 Nc
|
| Colors of variables: wff setvar class |
| Syntax hints: wb 176
wa 358
w3a 934
wex 1541
wceq 1642
wcel 1710
cvv 2859
1
cpw1 4135 class class class wbr 4639
(class class class)co 5525 cmap 5999 cen 6028 NC cncs 6088
Nc cnc 6091 ↑c cce 6096 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
| This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-si 4728 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 df-fv 4795 df-2nd 4797 df-ov 5526 df-oprab 5528 df-mpt 5652 df-mpt2 5654 df-txp 5736 df-compose 5748 df-ins2 5750 df-ins3 5752 df-image 5754 df-ins4 5756 df-si3 5758 df-funs 5760 df-fns 5762 df-pw1fn 5766 df-trans 5899 df-sym 5908 df-er 5909 df-ec 5947 df-qs 5951 df-map 6001 df-en 6029 df-ncs 6098 df-nc 6101 df-ce 6106 |
| This theorem is referenced by: ce0nnulb 6182 ceclb 6183 ce0 6190 ce2 6192 |
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