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Mirrors > Home > NFE Home > Th. List > ce2 | Unicode version |
Description: The value of base two cardinal exponentiation. Theorem XI.2.70 of [Rosser] p. 389. (Contributed by SF, 3-Mar-2015.) |
Ref | Expression |
---|---|
ce2.1 |
Ref | Expression |
---|---|
ce2 | Nc 1 2c ↑c Nc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5531 | . 2 Nc 1 2c ↑c 2c ↑c Nc 1 | |
2 | df-pr 3742 | . . . . . . . . . 10 | |
3 | pw1eq 4143 | . . . . . . . . . 10 1 1 | |
4 | 2, 3 | ax-mp 8 | . . . . . . . . 9 1 1 |
5 | pw1un 4163 | . . . . . . . . 9 1 1 1 | |
6 | 4, 5 | eqtri 2373 | . . . . . . . 8 1 1 1 |
7 | df-pr 3742 | . . . . . . . . 9 | |
8 | vvex 4109 | . . . . . . . . . . 11 | |
9 | 8 | pw1sn 4165 | . . . . . . . . . 10 1 |
10 | 0ex 4110 | . . . . . . . . . . 11 | |
11 | 10 | pw1sn 4165 | . . . . . . . . . 10 1 |
12 | 9, 11 | uneq12i 3416 | . . . . . . . . 9 1 1 |
13 | 7, 12 | eqtr4i 2376 | . . . . . . . 8 1 1 |
14 | 6, 13 | eqtr4i 2376 | . . . . . . 7 1 |
15 | vn0 3557 | . . . . . . . . . 10 | |
16 | 8 | sneqb 3876 | . . . . . . . . . . 11 |
17 | 16 | necon3bii 2548 | . . . . . . . . . 10 |
18 | 15, 17 | mpbir 200 | . . . . . . . . 9 |
19 | eqid 2353 | . . . . . . . . 9 | |
20 | snex 4111 | . . . . . . . . . 10 | |
21 | snex 4111 | . . . . . . . . . 10 | |
22 | neeq1 2524 | . . . . . . . . . . . 12 | |
23 | neeq2 2525 | . . . . . . . . . . . 12 | |
24 | 22, 23 | sylan9bb 680 | . . . . . . . . . . 11 |
25 | preq12 3801 | . . . . . . . . . . . 12 | |
26 | 25 | eqeq2d 2364 | . . . . . . . . . . 11 |
27 | 24, 26 | anbi12d 691 | . . . . . . . . . 10 |
28 | 20, 21, 27 | spc2ev 2947 | . . . . . . . . 9 |
29 | 18, 19, 28 | mp2an 653 | . . . . . . . 8 |
30 | el2c 6191 | . . . . . . . 8 2c | |
31 | 29, 30 | mpbir 200 | . . . . . . 7 2c |
32 | 14, 31 | eqeltri 2423 | . . . . . 6 1 2c |
33 | 2nc 6168 | . . . . . . 7 2c NC | |
34 | ncseqnc 6128 | . . . . . . 7 2c NC 2c Nc 1 1 2c | |
35 | 33, 34 | ax-mp 8 | . . . . . 6 2c Nc 1 1 2c |
36 | 32, 35 | mpbir 200 | . . . . 5 2c Nc 1 |
37 | 36 | oveq1i 5533 | . . . 4 2c ↑c Nc 1 Nc 1 ↑c Nc 1 |
38 | prex 4112 | . . . . 5 | |
39 | ce2.1 | . . . . 5 | |
40 | 38, 39 | cenc 6181 | . . . 4 Nc 1 ↑c Nc 1 Nc |
41 | 37, 40 | eqtri 2373 | . . 3 2c ↑c Nc 1 Nc |
42 | eqid 2353 | . . . . 5 | |
43 | 8, 10, 39 | enprmapc 6083 | . . . . 5 |
44 | 15, 42, 43 | mp2an 653 | . . . 4 |
45 | ovex 5551 | . . . . 5 | |
46 | 45 | eqnc 6127 | . . . 4 Nc Nc |
47 | 44, 46 | mpbir 200 | . . 3 Nc Nc |
48 | 41, 47 | eqtri 2373 | . 2 2c ↑c Nc 1 Nc |
49 | 1, 48 | syl6eq 2401 | 1 Nc 1 2c ↑c Nc |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 wex 1541 wceq 1642 wcel 1710 wne 2516 cvv 2859 cun 3207 c0 3550 cpw 3722 csn 3737 cpr 3738 1 cpw1 4135 class class class wbr 4639 (class class class)co 5525 cmap 5999 cen 6028 NC cncs 6088 Nc cnc 6091 2cc2c 6094 ↑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-2c 6104 df-ce 6106 |
This theorem is referenced by: ce2nc1 6193 ce2ncpw11c 6194 ce2lt 6220 ce2le 6233 tce2 6236 |
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