ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  enm Unicode version

Theorem enm 6516
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 6444 . . . . 5  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
2 f1of 5237 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  f : A
--> B )
3 ffvelrn 5416 . . . . . . . . 9  |-  ( ( f : A --> B  /\  x  e.  A )  ->  ( f `  x
)  e.  B )
4 elex2 2635 . . . . . . . . 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 113 . . . . . . 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 1534 . . . . 5  |-  ( E. f  f : A -1-1-onto-> B  ->  ( x  e.  A  ->  E. y  y  e.  B ) )
91, 8sylbi 119 . . . 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 1534 . 2  |-  ( E. x  x  e.  A  ->  ( A  ~~  B  ->  E. y  y  e.  B ) )
1211impcom 123 1  |-  ( ( A  ~~  B  /\  E. x  x  e.  A
)  ->  E. y 
y  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   E.wex 1426    e. wcel 1438   class class class wbr 3837   -->wf 4998   -1-1-onto->wf1o 5001   ` cfv 5002    ~~ cen 6435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-en 6438
This theorem is referenced by:  ssfilem  6571  diffitest  6583
  Copyright terms: Public domain W3C validator