Theorem canth2g 4492

Description: Cantor's theorem with the sethood requirement expressed as an antecedent. Theorem 23 of [Suppes] p. 97.

Assertion

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

Proof of Theorem canth2g

Step	Hyp	Ref	Expression
1		pweq 2408	. . 3 |- (x = A -> P~x = P~A)
2		breq12 2630	. . 3 |- ((x = A /\ P~x = P~A) -> (x ~< P~x <-> A ~< P~A))
3	1, 2	mpdan 706	. 2 |- (x = A -> (x ~< P~x <-> A ~< P~A))
4		visset 1816	. . 3 |- x e. V
5	4	canth2 4491	. 2 |- x ~< P~x
6	3, 5	vtoclg 1850	1 |- (A e. B -> A ~< P~A)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  P~cpw 2406   class class class wbr 2625   ~< csdm 4373

This theorem is referenced by:  pwuninel 4493  2pwuninel 4494  pwfi 4586  pwfiOLD 4587  canth3 4868  ondomon 4874

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2699  ax-sep 2709  ax-pow 2749  ax-pr 2786  ax-un 2873

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-pw 2407  df-sn 2417  df-pr 2418  df-op 2421  df-uni 2509  df-br 2626  df-opab 2673  df-id 2842  df-xp 3191  df-rel 3192  df-cnv 3193  df-co 3194  df-dm 3195  df-rn 3196  df-res 3197  df-ima 3198  df-fun 3199  df-fn 3200  df-f 3201  df-f1 3202  df-fo 3203  df-f1o 3204  df-fv 3205  df-en 4375  df-dom 4376  df-sdom 4377

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