MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  onpwsuc Structured version   Unicode version

Theorem onpwsuc 6530
Description: The collection of ordinal numbers in the power set of an ordinal number is its successor. (Contributed by NM, 19-Oct-2004.)
Assertion
Ref Expression
onpwsuc  |-  ( A  e.  On  ->  ( ~P A  i^i  On )  =  suc  A )

Proof of Theorem onpwsuc
StepHypRef Expression
1 eloni 4830 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ordpwsuc 6529 . 2  |-  ( Ord 
A  ->  ( ~P A  i^i  On )  =  suc  A )
31, 2syl 16 1  |-  ( A  e.  On  ->  ( ~P A  i^i  On )  =  suc  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    i^i cin 3428   ~Pcpw 3961   Ord word 4819   Oncon0 4820   suc csuc 4822
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 4514  ax-nul 4522  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-tr 4487  df-eprel 4733  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-suc 4826
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator