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Theorem bj-xpnzex 31064
Description: If the first factor of a product is nonempty, and the product is a set, then the second factor is a set. UPDATE: this is actually the exported form (curried form) of xpexcnv 6678 (up to commutation in the product). (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-xpnzex  |-  ( A  =/=  (/)  ->  ( ( A  X.  B )  e.  V  ->  B  e.  _V ) )

Proof of Theorem bj-xpnzex
StepHypRef Expression
1 0ex 4523 . . . . 5  |-  (/)  e.  _V
2 eleq1a 2483 . . . . 5  |-  ( (/)  e.  _V  ->  ( B  =  (/)  ->  B  e.  _V ) )
31, 2ax-mp 5 . . . 4  |-  ( B  =  (/)  ->  B  e. 
_V )
43a1d 25 . . 3  |-  ( B  =  (/)  ->  ( ( A  X.  B )  e.  V  ->  B  e.  _V ) )
54a1d 25 . 2  |-  ( B  =  (/)  ->  ( A  =/=  (/)  ->  ( ( A  X.  B )  e.  V  ->  B  e.  _V ) ) )
6 xpnz 5363 . . . 4  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( A  X.  B )  =/=  (/) )
7 xpexr2 6677 . . . . . 6  |-  ( ( ( A  X.  B
)  e.  V  /\  ( A  X.  B
)  =/=  (/) )  -> 
( A  e.  _V  /\  B  e.  _V )
)
87simprd 461 . . . . 5  |-  ( ( ( A  X.  B
)  e.  V  /\  ( A  X.  B
)  =/=  (/) )  ->  B  e.  _V )
98expcom 433 . . . 4  |-  ( ( A  X.  B )  =/=  (/)  ->  ( ( A  X.  B )  e.  V  ->  B  e.  _V ) )
106, 9sylbi 195 . . 3  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  ->  (
( A  X.  B
)  e.  V  ->  B  e.  _V )
)
1110expcom 433 . 2  |-  ( B  =/=  (/)  ->  ( A  =/=  (/)  ->  ( ( A  X.  B )  e.  V  ->  B  e.  _V ) ) )
125, 11pm2.61ine 2714 1  |-  ( A  =/=  (/)  ->  ( ( A  X.  B )  e.  V  ->  B  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840    =/= wne 2596   _Vcvv 3056   (/)c0 3735    X. cxp 4938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-xp 4946  df-rel 4947  df-cnv 4948  df-dm 4950  df-rn 4951
This theorem is referenced by:  bj-xpnzexb  31066
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