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Theorem xp1en 7664
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 7202 . . 3  |-  1o  =  { (/) }
21xpeq2i 4875 . 2  |-  ( A  X.  1o )  =  ( A  X.  { (/)
} )
3 0ex 4557 . . 3  |-  (/)  e.  _V
4 xpsneng 7663 . . 3  |-  ( ( A  e.  V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
53, 4mpan2 675 . 2  |-  ( A  e.  V  ->  ( A  X.  { (/) } ) 
~~  A )
62, 5syl5eqbr 4459 1  |-  ( A  e.  V  ->  ( A  X.  1o )  ~~  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1870   _Vcvv 3087   (/)c0 3767   {csn 4002   class class class wbr 4426    X. cxp 4852   1oc1o 7183    ~~ cen 7574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-int 4259  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-suc 5448  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-1o 7190  df-en 7578
This theorem is referenced by: (None)
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