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Mirrors > Home > NFE Home > Th. List > ce0lenc1 | Unicode version |
Description: Cardinal exponentiation to zero is a cardinal iff the number is less than the size of cardinal one. (Contributed by SF, 18-Mar-2015.) |
Ref | Expression |
---|---|
ce0lenc1 | NC ↑c 0c NC c Nc 1c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ce0tb 6238 | . 2 NC ↑c 0c NC NC Tc | |
2 | elncs 6119 | . . . . . 6 NC Nc | |
3 | tceq 6158 | . . . . . . . . 9 Nc Tc Tc Nc | |
4 | vex 2862 | . . . . . . . . . 10 | |
5 | 4 | tcnc 6225 | . . . . . . . . 9 Tc Nc Nc 1 |
6 | 3, 5 | syl6eq 2401 | . . . . . . . 8 Nc Tc Nc 1 |
7 | pw1ss1c 4158 | . . . . . . . . 9 1 1c | |
8 | 4 | pw1ex 4303 | . . . . . . . . . 10 1 |
9 | 1cex 4142 | . . . . . . . . . 10 1c | |
10 | 8, 9 | nclec 6195 | . . . . . . . . 9 1 1c Nc 1 c Nc 1c |
11 | 7, 10 | ax-mp 5 | . . . . . . . 8 Nc 1 c Nc 1c |
12 | 6, 11 | syl6eqbr 4676 | . . . . . . 7 Nc Tc c Nc 1c |
13 | 12 | exlimiv 1634 | . . . . . 6 Nc Tc c Nc 1c |
14 | 2, 13 | sylbi 187 | . . . . 5 NC Tc c Nc 1c |
15 | breq1 4642 | . . . . 5 Tc c Nc 1c Tc c Nc 1c | |
16 | 14, 15 | syl5ibrcom 213 | . . . 4 NC Tc c Nc 1c |
17 | 16 | rexlimiv 2732 | . . 3 NC Tc c Nc 1c |
18 | 9 | lenc 6223 | . . . 4 NC c Nc 1c 1c |
19 | ncseqnc 6128 | . . . . . . 7 NC Nc | |
20 | 19 | biimpar 471 | . . . . . 6 NC Nc |
21 | 4 | sspw12 4336 | . . . . . . . 8 1c 1 |
22 | vex 2862 | . . . . . . . . . . . 12 | |
23 | 22 | ncelncsi 6121 | . . . . . . . . . . 11 Nc NC |
24 | 22 | tcnc 6225 | . . . . . . . . . . 11 Tc Nc Nc 1 |
25 | tceq 6158 | . . . . . . . . . . . . 13 Nc Tc Tc Nc | |
26 | 25 | eqeq1d 2361 | . . . . . . . . . . . 12 Nc Tc Nc 1 Tc Nc Nc 1 |
27 | 26 | rspcev 2955 | . . . . . . . . . . 11 Nc NC Tc Nc Nc 1 NC Tc Nc 1 |
28 | 23, 24, 27 | mp2an 653 | . . . . . . . . . 10 NC Tc Nc 1 |
29 | nceq 6108 | . . . . . . . . . . . . 13 1 Nc Nc 1 | |
30 | 29 | eqeq1d 2361 | . . . . . . . . . . . 12 1 Nc Tc Nc 1 Tc |
31 | eqcom 2355 | . . . . . . . . . . . 12 Nc 1 Tc Tc Nc 1 | |
32 | 30, 31 | syl6bb 252 | . . . . . . . . . . 11 1 Nc Tc Tc Nc 1 |
33 | 32 | rexbidv 2635 | . . . . . . . . . 10 1 NC Nc Tc NC Tc Nc 1 |
34 | 28, 33 | mpbiri 224 | . . . . . . . . 9 1 NC Nc Tc |
35 | 34 | exlimiv 1634 | . . . . . . . 8 1 NC Nc Tc |
36 | 21, 35 | sylbi 187 | . . . . . . 7 1c NC Nc Tc |
37 | eqeq1 2359 | . . . . . . . 8 Nc Tc Nc Tc | |
38 | 37 | rexbidv 2635 | . . . . . . 7 Nc NC Tc NC Nc Tc |
39 | 36, 38 | syl5ibr 212 | . . . . . 6 Nc 1c NC Tc |
40 | 20, 39 | syl 15 | . . . . 5 NC 1c NC Tc |
41 | 40 | rexlimdva 2738 | . . . 4 NC 1c NC Tc |
42 | 18, 41 | sylbid 206 | . . 3 NC c Nc 1c NC Tc |
43 | 17, 42 | impbid2 195 | . 2 NC NC Tc c Nc 1c |
44 | 1, 43 | bitrd 244 | 1 NC ↑c 0c NC c Nc 1c |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 wex 1541 wceq 1642 wcel 1710 wrex 2615 wss 3257 1cc1c 4134 1 cpw1 4135 0cc0c 4374 class class class wbr 4639 (class class class)co 5525 NC cncs 6088 c clec 6089 Nc cnc 6091 Tc ctc 6093 ↑c cce 6096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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-lec 6099 df-nc 6101 df-tc 6103 df-ce 6106 |
This theorem is referenced by: nchoicelem8 6296 nchoicelem9 6297 |
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