MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xp1en Structured version   Unicode version

Theorem xp1en 7600
Description: One times a cardinal number. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
xp1en  |-  ( A  e.  V  ->  ( A  X.  1o )  ~~  A )

Proof of Theorem xp1en
StepHypRef Expression
1 df1o2 7139 . . 3  |-  1o  =  { (/) }
21xpeq2i 5020 . 2  |-  ( A  X.  1o )  =  ( A  X.  { (/)
} )
3 0ex 4577 . . 3  |-  (/)  e.  _V
4 xpsneng 7599 . . 3  |-  ( ( A  e.  V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
53, 4mpan2 671 . 2  |-  ( A  e.  V  ->  ( A  X.  { (/) } ) 
~~  A )
62, 5syl5eqbr 4480 1  |-  ( A  e.  V  ->  ( A  X.  1o )  ~~  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   _Vcvv 3113   (/)c0 3785   {csn 4027   class class class wbr 4447    X. cxp 4997   1oc1o 7120    ~~ cen 7510
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-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-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-1o 7127  df-en 7514
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator