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Mirrors > Home > NFE Home > Th. List > tfinprop | Unicode version |
Description: Properties of the finite T operator for a non-empty natural. Theorem X.1.28 of [Rosser] p. 528. (Contributed by SF, 22-Jan-2015.) |
Ref | Expression |
---|---|
tfinprop | Nn Tfin Nn 1 Tfin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tfin 4443 | . . 3 Tfin Nn 1 | |
2 | df-ne 2518 | . . . . . 6 | |
3 | iffalse 3669 | . . . . . 6 Nn 1 Nn 1 | |
4 | 2, 3 | sylbi 187 | . . . . 5 Nn 1 Nn 1 |
5 | 4 | adantl 452 | . . . 4 Nn Nn 1 Nn 1 |
6 | nnpw1ex 4484 | . . . . 5 Nn Nn 1 | |
7 | reiotacl 4364 | . . . . 5 Nn 1 Nn 1 Nn | |
8 | 6, 7 | syl 15 | . . . 4 Nn Nn 1 Nn |
9 | 5, 8 | eqeltrd 2427 | . . 3 Nn Nn 1 Nn |
10 | 1, 9 | syl5eqel 2437 | . 2 Nn Tfin Nn |
11 | 1, 5 | syl5req 2398 | . . 3 Nn Nn 1 Tfin |
12 | 10, 6 | jca 518 | . . . 4 Nn Tfin Nn Nn 1 |
13 | eleq2 2414 | . . . . . 6 Tfin 1 1 Tfin | |
14 | 13 | rexbidv 2635 | . . . . 5 Tfin 1 1 Tfin |
15 | 14 | reiota2 4368 | . . . 4 Tfin Nn Nn 1 1 Tfin Nn 1 Tfin |
16 | 12, 15 | syl 15 | . . 3 Nn 1 Tfin Nn 1 Tfin |
17 | 11, 16 | mpbird 223 | . 2 Nn 1 Tfin |
18 | 10, 17 | jca 518 | 1 Nn Tfin Nn 1 Tfin |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 176 wa 358 wceq 1642 wcel 1710 wne 2516 wrex 2615 wreu 2616 c0 3550 cif 3662 1 cpw1 4135 cio 4337 Nn cnnc 4373 Tfin ctfin 4435 |
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-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: tfinnnul 4490 tfincl 4492 tfin11 4493 tfinpw1 4494 tfinltfinlem1 4500 tfinltfin 4501 eventfin 4517 oddtfin 4518 sfinltfin 4535 vfinncvntnn 4548 |
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