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

Theorem enm 6682
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 6609 . . . . 5  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
2 f1of 5335 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  f : A
--> B )
3 ffvelrn 5521 . . . . . . . . 9  |-  ( ( f : A --> B  /\  x  e.  A )  ->  ( f `  x
)  e.  B )
4 elex2 2676 . . . . . . . . 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 1562 . . . . 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 1562 . 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 1453    e. wcel 1465   class class class wbr 3899   -->wf 5089   -1-1-onto->wf1o 5092   ` cfv 5093    ~~ cen 6600
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-en 6603
This theorem is referenced by:  ssfilem  6737  diffitest  6749
  Copyright terms: Public domain W3C validator