Theorem tfinpw1 4494

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.)

Assertion

Ref	Expression
tfinpw1	|- ((M e. Nn /\ A e. M) -> ~P1 A e. Tfin M)

Proof of Theorem tfinpw1

Dummy variables b n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ne0i 3556	. . 3 |- (A e. M -> M =/= (/))
2		tfinprop 4489	. . 3 |- ((M e. Nn /\ M =/= (/)) -> ( Tfin M e. Nn /\ E.b e. M ~P1 b e. Tfin M))
3	1, 2	sylan2 460	. 2 |- ((M e. Nn /\ A e. M) -> ( Tfin M e. Nn /\ E.b e. M ~P1 b e. Tfin M))
4		ncfinraise 4481	. . . . . . . 8 |- ((M e. Nn /\ A e. M /\ b e. M) -> E.n e. Nn (~P1 A e. n /\ ~P1 b e. n))
5	4	3expa 1151	. . . . . . 7 |- (((M e. Nn /\ A e. M) /\ b e. M) -> E.n e. Nn (~P1 A e. n /\ ~P1 b e. n))
6	5	adantrr 697	. . . . . 6 |- (((M e. Nn /\ A e. M) /\ (b e. M /\ ~P1 b e. Tfin M)) -> E.n e. Nn (~P1 A e. n /\ ~P1 b e. n))
7		simp3rl 1028	. . . . . . . . . 10 |- (((M e. Nn /\ A e. M) /\ (b e. M /\ ~P1 b e. Tfin M) /\ (n e. Nn /\ (~P1 A e. n /\ ~P1 b e. n))) -> ~P1 A e. n)
8		simp3l 983	. . . . . . . . . . 11 |- (((M e. Nn /\ A e. M) /\ (b e. M /\ ~P1 b e. Tfin M) /\ (n e. Nn /\ (~P1 A e. n /\ ~P1 b e. n))) -> n e. Nn )
9		simp1l 979	. . . . . . . . . . . 12 |- (((M e. Nn /\ A e. M) /\ (b e. M /\ ~P1 b e. Tfin M) /\ (n e. Nn /\ (~P1 A e. n /\ ~P1 b e. n))) -> M e. Nn )
10		tfincl 4492	. . . . . . . . . . . 12 |- (M e. Nn -> Tfin M e. Nn )
11	9, 10	syl 15	. . . . . . . . . . 11 |- (((M e. Nn /\ A e. M) /\ (b e. M /\ ~P1 b e. Tfin M) /\ (n e. Nn /\ (~P1 A e. n /\ ~P1 b e. n))) -> Tfin M e. Nn )
12		simp3rr 1029	. . . . . . . . . . 11 |- (((M e. Nn /\ A e. M) /\ (b e. M /\ ~P1 b e. Tfin M) /\ (n e. Nn /\ (~P1 A e. n /\ ~P1 b e. n))) -> ~P1 b e. n)
13		simp2r 982	. . . . . . . . . . 11 |- (((M e. Nn /\ A e. M) /\ (b e. M /\ ~P1 b e. Tfin M) /\ (n e. Nn /\ (~P1 A e. n /\ ~P1 b e. n))) -> ~P1 b e. Tfin M)
14		nnceleq 4430	. . . . . . . . . . 11 |- (((n e. Nn /\ Tfin M e. Nn ) /\ (~P1 b e. n /\ ~P1 b e. Tfin M)) -> n = Tfin M)
15	8, 11, 12, 13, 14	syl22anc 1183	. . . . . . . . . 10 |- (((M e. Nn /\ A e. M) /\ (b e. M /\ ~P1 b e. Tfin M) /\ (n e. Nn /\ (~P1 A e. n /\ ~P1 b e. n))) -> n = Tfin M)
16	7, 15	eleqtrd 2429	. . . . . . . . 9 |- (((M e. Nn /\ A e. M) /\ (b e. M /\ ~P1 b e. Tfin M) /\ (n e. Nn /\ (~P1 A e. n /\ ~P1 b e. n))) -> ~P1 A e. Tfin M)
17	16	3expa 1151	. . . . . . . 8 |- ((((M e. Nn /\ A e. M) /\ (b e. M /\ ~P1 b e. Tfin M)) /\ (n e. Nn /\ (~P1 A e. n /\ ~P1 b e. n))) -> ~P1 A e. Tfin M)
18	17	expr 598	. . . . . . 7 |- ((((M e. Nn /\ A e. M) /\ (b e. M /\ ~P1 b e. Tfin M)) /\ n e. Nn ) -> ((~P1 A e. n /\ ~P1 b e. n) -> ~P1 A e. Tfin M))
19	18	rexlimdva 2738	. . . . . 6 |- (((M e. Nn /\ A e. M) /\ (b e. M /\ ~P1 b e. Tfin M)) -> (E.n e. Nn (~P1 A e. n /\ ~P1 b e. n) -> ~P1 A e. Tfin M))
20	6, 19	mpd 14	. . . . 5 |- (((M e. Nn /\ A e. M) /\ (b e. M /\ ~P1 b e. Tfin M)) -> ~P1 A e. Tfin M)
21	20	expr 598	. . . 4 |- (((M e. Nn /\ A e. M) /\ b e. M) -> (~P1 b e. Tfin M -> ~P1 A e. Tfin M))
22	21	rexlimdva 2738	. . 3 |- ((M e. Nn /\ A e. M) -> (E.b e. M ~P1 b e. Tfin M -> ~P1 A e. Tfin M))
23	22	adantld 453	. 2 |- ((M e. Nn /\ A e. M) -> (( Tfin M e. Nn /\ E.b e. M ~P1 b e. Tfin M) -> ~P1 A e. Tfin M))
24	3, 23	mpd 14	1 |- ((M e. Nn /\ A e. M) -> ~P1 A e. Tfin M)

Syntax hints:   -> wi 4   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710   =/= wne 2516  E.wrex 2615  (/)c0 3550  ~P₁ cpw1 4135   Nn cnnc 4373   T_fin 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