Theorem pw1fin 6170

Description: The unit power class of a finite set is finite. (Contributed by SF, 3-Mar-2015.)

Assertion

Ref	Expression
pw1fin	|- (A e. Fin -> ~P1 A e. Fin )

Proof of Theorem pw1fin

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

Step	Hyp	Ref	Expression
1		ncfinraise 4482	. . . . 5 |- ((n e. Nn /\ A e. n /\ A e. n) -> E.m e. Nn (~P1 A e. m /\ ~P1 A e. m))
2	1	3anidm23 1241	. . . 4 |- ((n e. Nn /\ A e. n) -> E.m e. Nn (~P1 A e. m /\ ~P1 A e. m))
3	2	rexlimiva 2734	. . 3 |- (E.n e. Nn A e. n -> E.m e. Nn (~P1 A e. m /\ ~P1 A e. m))
4		simpl 443	. . . 4 |- ((~P1 A e. m /\ ~P1 A e. m) -> ~P1 A e. m)
5	4	reximi 2722	. . 3 |- (E.m e. Nn (~P1 A e. m /\ ~P1 A e. m) -> E.m e. Nn ~P1 A e. m)
6	3, 5	syl 15	. 2 |- (E.n e. Nn A e. n -> E.m e. Nn ~P1 A e. m)
7		elfin 4421	. 2 |- (A e. Fin <-> E.n e. Nn A e. n)
8		elfin 4421	. 2 |- (~P1 A e. Fin <-> E.m e. Nn ~P1 A e. m)
9	6, 7, 8	3imtr4i 257	1 |- (A e. Fin -> ~P1 A e. Fin )

Syntax hints:   -> wi 4   /\ wa 358   e. wcel 1710  E.wrex 2616  ~P₁ cpw1 4136   Nn cnnc 4374   Fin cfin 4377

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381

This theorem is referenced by:  nntccl  6171