Theorem canth2g 7668

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 4013	. . 3 |- ( x = A -> ~P x = ~P A )
2		breq12 4452	. . 3 |- ( ( x = A /\ ~P x = ~P A ) -> ( x ~< ~P x <-> A ~< ~P A ) )
3	1, 2	mpdan 668	. 2 |- ( x = A -> ( x ~< ~P x <-> A ~< ~P A ) )
4		vex 3116	. . 3 |- x e. _V
5	4	canth2 7667	. 2 |- x ~< ~P x
6	3, 5	vtoclg 3171	1 |- ( A e. V -> A ~< ~P A )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1379   e. wcel 1767   ~Pcpw 4010   class class class wbr 4447   ~< csdm 7512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-en 7514  df-dom 7515  df-sdom 7516

This theorem is referenced by:  2pwuninel  7669  2pwne  7670  pwfi  7811  cdalepw  8572  isfin32i  8741  fin34  8766  hsmexlem1  8802  canth3  8932  ondomon  8934  gchdomtri  9003  canthp1lem1  9026  canthp1lem2  9027  pwfseqlem5  9037  gchcdaidm  9042  gchxpidm  9043  gchpwdom  9044  gchaclem  9052  gchhar  9053  tsksdom  9130