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Theorem map0 6762
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 6761 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( A  ^m  B )  =  (/)  <->  ( A  =  (/)  /\  B  =/=  (/) ) ) )
41, 2, 3mp2an 656 1  |-  ( ( A  ^m  B )  =  (/)  <->  ( A  =  (/)  /\  B  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   _Vcvv 2757   (/)c0 3416  (class class class)co 5778    ^m cmap 6726
This theorem is referenced by:  psrbas  16072  birthdaylem1  20194
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-map 6728
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