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Theorem xpnz 5275
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 3753 . . . . 5  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 n0 3753 . . . . 5  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
31, 2anbi12i 708 . . . 4  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( E. x  x  e.  A  /\  E. y  y  e.  B ) )
4 eeanv 2089 . . . 4  |-  ( E. x E. y ( x  e.  A  /\  y  e.  B )  <->  ( E. x  x  e.  A  /\  E. y 
y  e.  B ) )
53, 4bitr4i 260 . . 3  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  E. x E. y ( x  e.  A  /\  y  e.  B ) )
6 opex 4678 . . . . . 6  |-  <. x ,  y >.  e.  _V
7 eleq1 2528 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( z  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
8 opelxp 4883 . . . . . . 7  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
97, 8syl6bb 269 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( z  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) ) )
106, 9spcev 3153 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  ->  E. z  z  e.  ( A  X.  B
) )
11 n0 3753 . . . . 5  |-  ( ( A  X.  B )  =/=  (/)  <->  E. z  z  e.  ( A  X.  B
) )
1210, 11sylibr 217 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( A  X.  B
)  =/=  (/) )
1312exlimivv 1789 . . 3  |-  ( E. x E. y ( x  e.  A  /\  y  e.  B )  ->  ( A  X.  B
)  =/=  (/) )
145, 13sylbi 200 . 2  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  ->  ( A  X.  B )  =/=  (/) )
15 xpeq1 4867 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  B )  =  ( (/)  X.  B
) )
16 0xp 4934 . . . . 5  |-  ( (/)  X.  B )  =  (/)
1715, 16syl6eq 2512 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  B )  =  (/) )
1817necon3i 2668 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  A  =/=  (/) )
19 xpeq2 4868 . . . . 5  |-  ( B  =  (/)  ->  ( A  X.  B )  =  ( A  X.  (/) ) )
20 xp0 5274 . . . . 5  |-  ( A  X.  (/) )  =  (/)
2119, 20syl6eq 2512 . . . 4  |-  ( B  =  (/)  ->  ( A  X.  B )  =  (/) )
2221necon3i 2668 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  B  =/=  (/) )
2318, 22jca 539 . 2  |-  ( ( A  X.  B )  =/=  (/)  ->  ( A  =/=  (/)  /\  B  =/=  (/) ) )
2414, 23impbii 192 1  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( A  X.  B )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    = wceq 1455   E.wex 1674    e. wcel 1898    =/= wne 2633   (/)c0 3743   <.cop 3986    X. cxp 4851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4417  df-opab 4476  df-xp 4859  df-rel 4860  df-cnv 4861
This theorem is referenced by:  xpeq0  5276  ssxpb  5290  xp11  5291  unixpid  5390  xpexr2  6761  frxp  6933  xpfir  7820  axcc2lem  8892  axdc4lem  8911  mamufacex  19463  txindis  20698  bj-xpnzex  31597  bj-1upln0  31648  bj-2upln1upl  31663  dibn0  34766
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