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Theorem enm 6680
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 6607 . . . . 5  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
2 f1of 5333 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  f : A
--> B )
3 ffvelrn 5519 . . . . . . . . 9  |-  ( ( f : A --> B  /\  x  e.  A )  ->  ( f `  x
)  e.  B )
4 elex2 2674 . . . . . . . . 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 114 . . . . . . 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 1560 . . . . 5  |-  ( E. f  f : A -1-1-onto-> B  ->  ( x  e.  A  ->  E. y  y  e.  B ) )
91, 8sylbi 120 . . . 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 1560 . 2  |-  ( E. x  x  e.  A  ->  ( A  ~~  B  ->  E. y  y  e.  B ) )
1211impcom 124 1  |-  ( ( A  ~~  B  /\  E. x  x  e.  A
)  ->  E. y 
y  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   E.wex 1451    e. wcel 1463   class class class wbr 3897   -->wf 5087   -1-1-onto->wf1o 5090   ` cfv 5091    ~~ cen 6598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-en 6601
This theorem is referenced by:  ssfilem  6735  diffitest  6747
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