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Theorem enm 6814
Description: A set equinumerous to an inhabited set is inhabited. (Contributed by Jim Kingdon, 19-May-2020.)
Assertion
Ref Expression
enm  |-  ( ( A  ~~  B  /\  E. x  x  e.  A
)  ->  E. y 
y  e.  B )
Distinct variable groups:    x, y, A   
x, B, y

Proof of Theorem enm
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 bren 6741 . . . . 5  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
2 f1of 5457 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  f : A
--> B )
3 ffvelcdm 5645 . . . . . . . . 9  |-  ( ( f : A --> B  /\  x  e.  A )  ->  ( f `  x
)  e.  B )
4 elex2 2753 . . . . . . . . 9  |-  ( ( f `  x )  e.  B  ->  E. y 
y  e.  B )
53, 4syl 14 . . . . . . . 8  |-  ( ( f : A --> B  /\  x  e.  A )  ->  E. y  y  e.  B )
65ex 115 . . . . . . 7  |-  ( f : A --> B  -> 
( x  e.  A  ->  E. y  y  e.  B ) )
72, 6syl 14 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  ( x  e.  A  ->  E. y 
y  e.  B ) )
87exlimiv 1598 . . . . 5  |-  ( E. f  f : A -1-1-onto-> B  ->  ( x  e.  A  ->  E. y  y  e.  B ) )
91, 8sylbi 121 . . . 4  |-  ( A 
~~  B  ->  (
x  e.  A  ->  E. y  y  e.  B ) )
109com12 30 . . 3  |-  ( x  e.  A  ->  ( A  ~~  B  ->  E. y 
y  e.  B ) )
1110exlimiv 1598 . 2  |-  ( E. x  x  e.  A  ->  ( A  ~~  B  ->  E. y  y  e.  B ) )
1211impcom 125 1  |-  ( ( A  ~~  B  /\  E. x  x  e.  A
)  ->  E. y 
y  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   E.wex 1492    e. wcel 2148   class class class wbr 4000   -->wf 5208   -1-1-onto->wf1o 5211   ` cfv 5212    ~~ cen 6732
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-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 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  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-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-en 6735
This theorem is referenced by:  ssfilem  6869  diffitest  6881
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