Theorem canth2g 8114

Description: Cantor's theorem with the sethood requirement expressed as an antecedent. Theorem 23 of [Suppes] p. 97. (Contributed by NM, 7-Nov-2003.)

Assertion

Ref	Expression
canth2g	|- ( A e. V -> A ~< ~P A )

Proof of Theorem canth2g

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 4161	. . 3 |- ( x = A -> ~P x = ~P A )
2		breq12 4658	. . 3 |- ( ( x = A /\ ~P x = ~P A ) -> ( x ~< ~P x <-> A ~< ~P A ) )
3	1, 2	mpdan 702	. 2 |- ( x = A -> ( x ~< ~P x <-> A ~< ~P A ) )
4		vex 3203	. . 3 |- x e. _V
5	4	canth2 8113	. 2 |- x ~< ~P x
6	3, 5	vtoclg 3266	1 |- ( A e. V -> A ~< ~P A )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   ~Pcpw 4158   class class class wbr 4653   ~< csdm 7954

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-pow 4843  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-csb 3534  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-en 7956  df-dom 7957  df-sdom 7958

This theorem is referenced by:  2pwuninel  8115  2pwne  8116  pwfi  8261  cdalepw  9018  isfin32i  9187  fin34  9212  hsmexlem1  9248  canth3  9383  ondomon  9385  gchdomtri  9451  canthp1lem1  9474  canthp1lem2  9475  pwfseqlem5  9485  gchcdaidm  9490  gchxpidm  9491  gchpwdom  9492  gchaclem  9500  gchhar  9501  tsksdom  9578