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Theorem xpnz 5260
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 3649 . . . . 5  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 n0 3649 . . . . 5  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
31, 2anbi12i 697 . . . 4  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( E. x  x  e.  A  /\  E. y  y  e.  B ) )
4 eeanv 1932 . . . 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 4559 . . . . . 6  |-  <. x ,  y >.  e.  _V
7 eleq1 2503 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( z  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
8 opelxp 4872 . . . . . . 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 3067 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  ->  E. z  z  e.  ( A  X.  B
) )
11 n0 3649 . . . . 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 1689 . . 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 4857 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  B )  =  ( (/)  X.  B
) )
16 0xp 4920 . . . . 5  |-  ( (/)  X.  B )  =  (/)
1715, 16syl6eq 2491 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  B )  =  (/) )
1817necon3i 2653 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  A  =/=  (/) )
19 xpeq2 4858 . . . . 5  |-  ( B  =  (/)  ->  ( A  X.  B )  =  ( A  X.  (/) ) )
20 xp0 5259 . . . . 5  |-  ( A  X.  (/) )  =  (/)
2119, 20syl6eq 2491 . . . 4  |-  ( B  =  (/)  ->  ( A  X.  B )  =  (/) )
2221necon3i 2653 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  B  =/=  (/) )
2318, 22jca 532 . 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 1369   E.wex 1586    e. wcel 1756    =/= wne 2609   (/)c0 3640   <.cop 3886    X. cxp 4841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pr 4534
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-rab 2727  df-v 2977  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-sn 3881  df-pr 3883  df-op 3887  df-br 4296  df-opab 4354  df-xp 4849  df-rel 4850  df-cnv 4851
This theorem is referenced by:  xpeq0  5261  ssxpb  5275  xp11  5276  unixpid  5375  xpexr2  6522  frxp  6685  xpfir  7538  axcc2lem  8608  axdc4lem  8627  mamufacex  18292  txindis  19210  bj-xpnzex  32454  bj-1upln0  32505  bj-2upln1upl  32520  dibn0  34801
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