Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-xpnzex Structured version   Unicode version

Theorem bj-xpnzex 33597
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 6723 (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 4577 . . . . 5  |-  (/)  e.  _V
2 eleq1a 2550 . . . . 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 5424 . . . 4  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( A  X.  B )  =/=  (/) )
7 xpexr2 6722 . . . . . 6  |-  ( ( ( A  X.  B
)  e.  V  /\  ( A  X.  B
)  =/=  (/) )  -> 
( A  e.  _V  /\  B  e.  _V )
)
87simprd 463 . . . . 5  |-  ( ( ( A  X.  B
)  e.  V  /\  ( A  X.  B
)  =/=  (/) )  ->  B  e.  _V )
98expcom 435 . . . 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 435 . 2  |-  ( B  =/=  (/)  ->  ( A  =/=  (/)  ->  ( ( A  X.  B )  e.  V  ->  B  e.  _V ) ) )
125, 11pm2.61ine 2780 1  |-  ( A  =/=  (/)  ->  ( ( A  X.  B )  e.  V  ->  B  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   (/)c0 3785    X. cxp 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010
This theorem is referenced by:  bj-xpnzexb  33599
  Copyright terms: Public domain W3C validator