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Theorem map0 6689
Description: Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.)
Hypotheses
Ref Expression
map0.1  |-  A  e. 
_V
map0.2  |-  B  e. 
_V
Assertion
Ref Expression
map0  |-  ( ( A  ^m  B )  =  (/)  <->  ( A  =  (/)  /\  B  =/=  (/) ) )

Proof of Theorem map0
StepHypRef Expression
1 map0.1 . 2  |-  A  e. 
_V
2 map0.2 . 2  |-  B  e. 
_V
3 map0g 6688 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( A  ^m  B )  =  (/)  <->  ( A  =  (/)  /\  B  =/=  (/) ) ) )
41, 2, 3mp2an 426 1  |-  ( ( A  ^m  B )  =  (/)  <->  ( A  =  (/)  /\  B  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148    =/= wne 2347   _Vcvv 2738   (/)c0 3423  (class class class)co 5875    ^m cmap 6648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-map 6650
This theorem is referenced by: (None)
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