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Theorem xpnz 5433
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 3803 . . . . 5  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 n0 3803 . . . . 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 1989 . . . 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 4720 . . . . . 6  |-  <. x ,  y >.  e.  _V
7 eleq1 2529 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( z  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
8 opelxp 5038 . . . . . . 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 3201 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  ->  E. z  z  e.  ( A  X.  B
) )
11 n0 3803 . . . . 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 1724 . . 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 5022 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  B )  =  ( (/)  X.  B
) )
16 0xp 5089 . . . . 5  |-  ( (/)  X.  B )  =  (/)
1715, 16syl6eq 2514 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  B )  =  (/) )
1817necon3i 2697 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  A  =/=  (/) )
19 xpeq2 5023 . . . . 5  |-  ( B  =  (/)  ->  ( A  X.  B )  =  ( A  X.  (/) ) )
20 xp0 5432 . . . . 5  |-  ( A  X.  (/) )  =  (/)
2119, 20syl6eq 2514 . . . 4  |-  ( B  =  (/)  ->  ( A  X.  B )  =  (/) )
2221necon3i 2697 . . 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 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   (/)c0 3793   <.cop 4038    X. cxp 5006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016
This theorem is referenced by:  xpeq0  5434  ssxpb  5448  xp11  5449  unixpid  5548  xpexr2  6740  frxp  6909  xpfir  7761  axcc2lem  8833  axdc4lem  8852  mamufacex  19018  txindis  20261  bj-xpnzex  34659  bj-1upln0  34710  bj-2upln1upl  34725  dibn0  37023
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