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Theorem 0xp 4935
Description: The Cartesian product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
0xp  |-  ( (/)  X.  A )  =  (/)

Proof of Theorem 0xp
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp 4871 . . 3  |-  ( z  e.  ( (/)  X.  A
)  <->  E. x E. y
( z  =  <. x ,  y >.  /\  (
x  e.  (/)  /\  y  e.  A ) ) )
2 noel 3771 . . . . . . 7  |-  -.  x  e.  (/)
3 simprl 762 . . . . . . 7  |-  ( ( z  =  <. x ,  y >.  /\  (
x  e.  (/)  /\  y  e.  A ) )  ->  x  e.  (/) )
42, 3mto 179 . . . . . 6  |-  -.  (
z  =  <. x ,  y >.  /\  (
x  e.  (/)  /\  y  e.  A ) )
54nex 1674 . . . . 5  |-  -.  E. y ( z  = 
<. x ,  y >.  /\  ( x  e.  (/)  /\  y  e.  A ) )
65nex 1674 . . . 4  |-  -.  E. x E. y ( z  =  <. x ,  y
>.  /\  ( x  e.  (/)  /\  y  e.  A
) )
7 noel 3771 . . . 4  |-  -.  z  e.  (/)
86, 72false 351 . . 3  |-  ( E. x E. y ( z  =  <. x ,  y >.  /\  (
x  e.  (/)  /\  y  e.  A ) )  <->  z  e.  (/) )
91, 8bitri 252 . 2  |-  ( z  e.  ( (/)  X.  A
)  <->  z  e.  (/) )
109eqriv 2425 1  |-  ( (/)  X.  A )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1870   (/)c0 3767   <.cop 4008    X. cxp 4852
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-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
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-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-opab 4485  df-xp 4860
This theorem is referenced by:  dmxpid  5074  csbres  5128  res0  5129  xp0  5275  xpnz  5276  xpdisj1  5278  difxp2  5283  xpcan2  5294  xpima  5299  unixp  5389  unixpid  5391  xpcoid  5397  fodomr  7729  xpfi  7848  cdaassen  8610  iundom2g  8963  alephadd  9000  hashxplem  12600  dmtrclfv  13061  ramcl  14950  0subcat  15698  mat0dimbas0  19426  mavmul0g  19513  txindislem  20583  txhaus  20597  tmdgsum  21045  ust0  21169  sibf0  29003  mexval2  29937  poimirlem5  31660  poimirlem10  31665  poimirlem22  31677  poimirlem23  31678  poimirlem26  31681  poimirlem28  31683  0mbf  31701  0heALT  36031
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