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| Mirrors > Home > NFE Home > Th. List > ce0nn | Unicode version | ||
| Description: A natural raised to cardinal zero is non-empty. Theorem XI.2.44 of [Rosser] p. 383. (Contributed by SF, 9-Mar-2015.) |
| Ref | Expression |
|---|---|
| ce0nn |
    Nn 
↑c 0c   |
are mutually distinct and distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vex 2862 |
. . . . . 6
 | |
| 2 | 1 | elcompl 3225 |
. . . . 5
∼       0c FullFun
↑c   0c FullFun ↑c |
| 3 | brres 4949 |
. . . . . . . . . 10
0c
![](wedge.gif "/\"){kind=small} 0c | |
| 4 | eliniseg 5020 |
. . . . . . . . . . 11
0c 0c | |
| 5 | 4 | anbi2i 675 |
. . . . . . . . . 10
0c
0c |
| 6 | 0cex 4392 |
. . . . . . . . . . 11
0c
| |
| 7 | 1, 6 | op1st2nd 5790 |
. . . . . . . . . 10
0c   
0c |
| 8 | 3, 5, 7 | 3bitri 262 |
. . . . . . . . 9
0c
0c |
| 9 | 8 | rexbii 2639 |
. . . . . . . 8
FullFun
↑c 0c
FullFun ↑c
0c |
| 10 | elima 4754 |
. . . . . . . 8
0c FullFun
↑c FullFun ↑c 0c | |
| 11 | risset 2661 |
. . . . . . . 8
0c FullFun
↑c
FullFun
↑c
0c | |
| 12 | 9, 10, 11 | 3bitr4i 268 |
. . . . . . 7
0c FullFun
↑c
0c FullFun
↑c |
| 13 | eliniseg 5020 |
. . . . . . 7
0c FullFun
↑c
0c FullFun ↑c | |
| 14 | 1, 6 | brfullfunop 5867 |
. . . . . . 7
0c FullFun ↑c ↑c
0c
|
| 15 | 12, 13, 14 | 3bitri 262 |
. . . . . 6
0c FullFun
↑c
↑c 0c |
| 16 | 15 | necon3bbii 2547 |
. . . . 5
0c FullFun ↑c ↑c
0c
|
| 17 | 2, 16 | bitri 240 |
. . . 4
∼ 0c FullFun
↑c
↑c 0c |
| 18 | 17 | abbi2i 2464 |
. . 3
∼ 0c FullFun ↑c 
↑c
0c
|
| 19 | 1stex 4739 |
. . . . . 6
| |
| 20 | 2ndex 5112 |
. . . . . . . 8
| |
| 21 | 20 | cnvex 5102 |
. . . . . . 7
|
| 22 | snex 4111 |
. . . . . . 7
0c | |
| 23 | 21, 22 | imaex 4747 |
. . . . . 6
0c |
| 24 | 19, 23 | resex 5117 |
. . . . 5
0c
|
| 25 | ceex 6174 |
. . . . . . . 8
↑c | |
| 26 | 25 | fullfunex 5860 |
. . . . . . 7
FullFun ↑c |
| 27 | 26 | cnvex 5102 |
. . . . . 6
FullFun ↑c |
| 28 | snex 4111 |
. . . . . 6
| |
| 29 | 27, 28 | imaex 4747 |
. . . . 5
FullFun
↑c |
| 30 | 24, 29 | imaex 4747 |
. . . 4
0c FullFun ↑c |
| 31 | 30 | complex 4104 |
. . 3
∼ 0c FullFun ↑c |
| 32 | 18, 31 | eqeltrri 2424 |
. 2
↑c 0c |
| 33 | oveq1 5530 |
. . 3
0c
↑c 0c 0c ↑c
0c | |
| 34 | 33 | neeq1d 2529 |
. 2
0c ↑c
0c
0c
↑c 0c |
| 35 | oveq1 5530 |
. . 3
↑c 0c
↑c 0c | |
| 36 | 35 | neeq1d 2529 |
. 2
↑c 0c
↑c
0c
|
| 37 | oveq1 5530 |
. . 3
 1c
↑c 0c 1c ↑c 0c | |
| 38 | 37 | neeq1d 2529 |
. 2
1c ↑c
0c
1c
↑c 0c |
| 39 | oveq1 5530 |
. . 3
↑c 0c
↑c 0c | |
| 40 | 39 | neeq1d 2529 |
. 2
↑c 0c
↑c
0c
|
| 41 | 0cnc 6138 |
. . 3
0c
NC | |
| 42 | pw10 4161 |
. . . 4
 1 | |
| 43 | nulel0c 4422 |
. . . 4
0c | |
| 44 | 42, 43 | eqeltri 2423 |
. . 3
1
0c |
| 45 | ce0nnuli 6178 |
. . 3
0c NC 1 0c 0c ↑c
0c
| |
| 46 | 41, 44, 45 | mp2an 653 |
. 2
0c
↑c 0c |
| 47 | nnnc 6146 |
. . 3
Nn NC | |
| 48 | 1cnc 6139 |
. . . . . 6
1c
NC | |
| 49 | 0ex 4110 |
. . . . . . . 8
| |
| 50 | 49 | pw1sn 4165 |
. . . . . . 7
1 |
| 51 | 28 | snel1c 4140 |
. . . . . . 7
1c |
| 52 | 50, 51 | eqeltri 2423 |
. . . . . 6
1 1c |
| 53 | ce0nnuli 6178 |
. . . . . 6
1c NC 1
1c
1c
↑c 0c | |
| 54 | 48, 52, 53 | mp2an 653 |
. . . . 5
1c
↑c 0c |
| 55 | 54 | jctr 526 |
. . . 4
↑c 0c ↑c 0c 1c
↑c 0c |
| 56 | ce0addcnnul 6179 |
. . . . 5
NC 1c NC
1c
↑c 0c
↑c 0c 1c ↑c
0c
| |
| 57 | 48, 56 | mpan2 652 |
. . . 4
NC 1c ↑c 0c ↑c
0c
1c
↑c 0c |
| 58 | 55, 57 | syl5ibr 212 |
. . 3
NC ↑c
0c
1c
↑c 0c |
| 59 | 47, 58 | syl 15 |
. 2
Nn ↑c
0c
1c
↑c 0c |
| 60 | 32, 34, 36, 38, 40, 46, 59 | finds 4411 |
1
Nn
↑c 0c |
| Colors of variables: wff setvar class |
| Syntax hints: wn 3
wi 4
wb 176
wa 358
wceq 1642
wcel 1710
cab 2339
wne 2516
wrex 2615
cvv 2859
∼ ccompl 3205 c0 3550 csn 3737 1cc1c 4134
1
cpw1 4135 Nn cnnc 4373
0cc0c 4374 cplc 4375 cop 4561 class class class wbr 4639
c1st 4717
cima 4722
ccnv 4771
cres 4774
c2nd 4783
(class class class)co 5525 FullFun
cfullfun 5767 NC cncs 6088
↑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-ins2 5750 df-ins3 5752 df-image 5754 df-ins4 5756 df-si3 5758 df-funs 5760 df-fns 5762 df-pw1fn 5766 df-fullfun 5768 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: ceclnn1 6189 nchoicelem5 6293 |
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