Theorem pw1fin 6169

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 4481	. . . . 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 2733	. . 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 2721	. . 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 4420	. 2 |- (A e. Fin <-> E.n e. Nn A e. n)
8		elfin 4420	. 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 2615  ~P₁ cpw1 4135   Nn cnnc 4373   Fin cfin 4376

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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  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-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380

This theorem is referenced by:  nntccl  6170