Theorem canthwdom 8484

Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 8113, equivalent to canth 6608). (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
canthwdom	|- -. ~P A ~<_* A

Proof of Theorem canthwdom

Dummy variables x f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0elpw 4834	. . . . 5 |- (/) e. ~P A
2		ne0i 3921	. . . . 5 |- ( (/) e. ~P A -> ~P A =/= (/) )
3	1, 2	mp1i 13	. . . 4 |- ( ~P A ~<_* A -> ~P A =/= (/) )
4		brwdomn0 8474	. . . 4 |- ( ~P A =/= (/) -> ( ~P A ~<_* A <-> E. f f : A -onto-> ~P A ) )
5	3, 4	syl 17	. . 3 |- ( ~P A ~<_* A -> ( ~P A ~<_* A <-> E. f f : A -onto-> ~P A ) )
6	5	ibi 256	. 2 |- ( ~P A ~<_* A -> E. f f : A -onto-> ~P A )
7		relwdom 8471	. . . . 5 |- Rel ~<_*
8	7	brrelex2i 5159	. . . 4 |- ( ~P A ~<_* A -> A e. _V )
9		foeq2 6112	. . . . . . 7 |- ( x = A -> ( f : x -onto-> ~P x <-> f : A -onto-> ~P x ) )
10		pweq 4161	. . . . . . . 8 |- ( x = A -> ~P x = ~P A )
11		foeq3 6113	. . . . . . . 8 |- ( ~P x = ~P A -> ( f : A -onto-> ~P x <-> f : A -onto-> ~P A ) )
12	10, 11	syl 17	. . . . . . 7 |- ( x = A -> ( f : A -onto-> ~P x <-> f : A -onto-> ~P A ) )
13	9, 12	bitrd 268	. . . . . 6 |- ( x = A -> ( f : x -onto-> ~P x <-> f : A -onto-> ~P A ) )
14	13	notbid 308	. . . . 5 |- ( x = A -> ( -. f : x -onto-> ~P x <-> -. f : A -onto-> ~P A ) )
15		vex 3203	. . . . . 6 |- x e. _V
16	15	canth 6608	. . . . 5 |- -. f : x -onto-> ~P x
17	14, 16	vtoclg 3266	. . . 4 |- ( A e. _V -> -. f : A -onto-> ~P A )
18	8, 17	syl 17	. . 3 |- ( ~P A ~<_* A -> -. f : A -onto-> ~P A )
19	18	nexdv 1864	. 2 |- ( ~P A ~<_* A -> -. E. f f : A -onto-> ~P A )
20	6, 19	pm2.65i 185	1 |- -. ~P A ~<_* A

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   -onto->wfo 5886   ~<_^* cwdom 8462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-wdom 8464

This theorem is referenced by:  pwcdadom  9038