Theorem canth2g 7567

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 3963	. . 3 |- ( x = A -> ~P x = ~P A )
2		breq12 4397	. . 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 3073	. . 3 |- x e. _V
5	4	canth2 7566	. 2 |- x ~< ~P x
6	3, 5	vtoclg 3128	1 |- ( A e. V -> A ~< ~P A )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1370   e. wcel 1758   ~Pcpw 3960   class class class wbr 4392   ~< csdm 7411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-en 7413  df-dom 7414  df-sdom 7415

This theorem is referenced by:  2pwuninel  7568  2pwne  7569  pwfi  7709  cdalepw  8468  isfin32i  8637  fin34  8662  hsmexlem1  8698  canth3  8828  ondomon  8830  gchdomtri  8899  canthp1lem1  8922  canthp1lem2  8923  pwfseqlem5  8933  gchcdaidm  8938  gchxpidm  8939  gchpwdom  8940  gchaclem  8948  gchhar  8949  tsksdom  9026