New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > tfinpw1 | Unicode version |
Description: The finite T operator on a natural contains the unit power class of any element of the natural. Theorem X.1.31 of [Rosser] p. 528. (Contributed by SF, 24-Jan-2015.) |
Ref | Expression |
---|---|
tfinpw1 | Nn 1 Tfin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0i 3556 | . . 3 | |
2 | tfinprop 4489 | . . 3 Nn Tfin Nn 1 Tfin | |
3 | 1, 2 | sylan2 460 | . 2 Nn Tfin Nn 1 Tfin |
4 | ncfinraise 4481 | . . . . . . . 8 Nn Nn 1 1 | |
5 | 4 | 3expa 1151 | . . . . . . 7 Nn Nn 1 1 |
6 | 5 | adantrr 697 | . . . . . 6 Nn 1 Tfin Nn 1 1 |
7 | simp3rl 1028 | . . . . . . . . . 10 Nn 1 Tfin Nn 1 1 1 | |
8 | simp3l 983 | . . . . . . . . . . 11 Nn 1 Tfin Nn 1 1 Nn | |
9 | simp1l 979 | . . . . . . . . . . . 12 Nn 1 Tfin Nn 1 1 Nn | |
10 | tfincl 4492 | . . . . . . . . . . . 12 Nn Tfin Nn | |
11 | 9, 10 | syl 15 | . . . . . . . . . . 11 Nn 1 Tfin Nn 1 1 Tfin Nn |
12 | simp3rr 1029 | . . . . . . . . . . 11 Nn 1 Tfin Nn 1 1 1 | |
13 | simp2r 982 | . . . . . . . . . . 11 Nn 1 Tfin Nn 1 1 1 Tfin | |
14 | nnceleq 4430 | . . . . . . . . . . 11 Nn Tfin Nn 1 1 Tfin Tfin | |
15 | 8, 11, 12, 13, 14 | syl22anc 1183 | . . . . . . . . . 10 Nn 1 Tfin Nn 1 1 Tfin |
16 | 7, 15 | eleqtrd 2429 | . . . . . . . . 9 Nn 1 Tfin Nn 1 1 1 Tfin |
17 | 16 | 3expa 1151 | . . . . . . . 8 Nn 1 Tfin Nn 1 1 1 Tfin |
18 | 17 | expr 598 | . . . . . . 7 Nn 1 Tfin Nn 1 1 1 Tfin |
19 | 18 | rexlimdva 2738 | . . . . . 6 Nn 1 Tfin Nn 1 1 1 Tfin |
20 | 6, 19 | mpd 14 | . . . . 5 Nn 1 Tfin 1 Tfin |
21 | 20 | expr 598 | . . . 4 Nn 1 Tfin 1 Tfin |
22 | 21 | rexlimdva 2738 | . . 3 Nn 1 Tfin 1 Tfin |
23 | 22 | adantld 453 | . 2 Nn Tfin Nn 1 Tfin 1 Tfin |
24 | 3, 23 | mpd 14 | 1 Nn 1 Tfin |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 358 w3a 934 wceq 1642 wcel 1710 wne 2516 wrex 2615 c0 3550 1 cpw1 4135 Nn cnnc 4373 Tfin ctfin 4435 |
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-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-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-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-tfin 4443 |
This theorem is referenced by: ncfintfin 4495 tfindi 4496 tfin0c 4497 tfinsuc 4498 sfintfin 4532 sfinltfin 4535 vfinspsslem1 4550 |
Copyright terms: Public domain | W3C validator |