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Theorem xpnz 5245
Description: The Cartesian product of nonempty classes is nonempty. (Variation of a theorem contributed by Raph Levien, 30-Jun-2006.) (Contributed by NM, 30-Jun-2006.)
Assertion
Ref Expression
xpnz  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( A  X.  B )  =/=  (/) )

Proof of Theorem xpnz
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 3634 . . . . 5  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 n0 3634 . . . . 5  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
31, 2anbi12i 690 . . . 4  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( E. x  x  e.  A  /\  E. y  y  e.  B ) )
4 eeanv 1930 . . . 4  |-  ( E. x E. y ( x  e.  A  /\  y  e.  B )  <->  ( E. x  x  e.  A  /\  E. y 
y  e.  B ) )
53, 4bitr4i 252 . . 3  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  E. x E. y ( x  e.  A  /\  y  e.  B ) )
6 opex 4544 . . . . . 6  |-  <. x ,  y >.  e.  _V
7 eleq1 2493 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( z  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
8 opelxp 4856 . . . . . . 7  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
97, 8syl6bb 261 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( z  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) ) )
106, 9spcev 3053 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  ->  E. z  z  e.  ( A  X.  B
) )
11 n0 3634 . . . . 5  |-  ( ( A  X.  B )  =/=  (/)  <->  E. z  z  e.  ( A  X.  B
) )
1210, 11sylibr 212 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( A  X.  B
)  =/=  (/) )
1312exlimivv 1688 . . 3  |-  ( E. x E. y ( x  e.  A  /\  y  e.  B )  ->  ( A  X.  B
)  =/=  (/) )
145, 13sylbi 195 . 2  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  ->  ( A  X.  B )  =/=  (/) )
15 xpeq1 4841 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  B )  =  ( (/)  X.  B
) )
16 0xp 4904 . . . . 5  |-  ( (/)  X.  B )  =  (/)
1715, 16syl6eq 2481 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  B )  =  (/) )
1817necon3i 2640 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  A  =/=  (/) )
19 xpeq2 4842 . . . . 5  |-  ( B  =  (/)  ->  ( A  X.  B )  =  ( A  X.  (/) ) )
20 xp0 5244 . . . . 5  |-  ( A  X.  (/) )  =  (/)
2119, 20syl6eq 2481 . . . 4  |-  ( B  =  (/)  ->  ( A  X.  B )  =  (/) )
2221necon3i 2640 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  B  =/=  (/) )
2318, 22jca 529 . 2  |-  ( ( A  X.  B )  =/=  (/)  ->  ( A  =/=  (/)  /\  B  =/=  (/) ) )
2414, 23impbii 188 1  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( A  X.  B )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1362   E.wex 1589    e. wcel 1755    =/= wne 2596   (/)c0 3625   <.cop 3871    X. cxp 4825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-br 4281  df-opab 4339  df-xp 4833  df-rel 4834  df-cnv 4835
This theorem is referenced by:  xpeq0  5246  ssxpb  5260  xp11  5261  unixpid  5360  xpexr2  6508  frxp  6671  xpfir  7523  axcc2lem  8593  axdc4lem  8612  mamufacex  18131  txindis  19049  bj-xpnzex  32055  bj-1upln0  32106  bj-2upln1upl  32121  dibn0  34392
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