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Theorem xp1en 7500
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 7035 . . 3  |-  1o  =  { (/) }
21xpeq2i 4962 . 2  |-  ( A  X.  1o )  =  ( A  X.  { (/)
} )
3 0ex 4523 . . 3  |-  (/)  e.  _V
4 xpsneng 7499 . . 3  |-  ( ( A  e.  V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
53, 4mpan2 671 . 2  |-  ( A  e.  V  ->  ( A  X.  { (/) } ) 
~~  A )
62, 5syl5eqbr 4426 1  |-  ( A  e.  V  ->  ( A  X.  1o )  ~~  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758   _Vcvv 3071   (/)c0 3738   {csn 3978   class class class wbr 4393    X. cxp 4939   1oc1o 7016    ~~ cen 7410
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-pow 4571  ax-pr 4632  ax-un 6475
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 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-int 4230  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-1o 7023  df-en 7414
This theorem is referenced by: (None)
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