Theorem en0 6937

Description: The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.)

Assertion

Ref	Expression
en0	|- ( A ~~ (/) <-> A = (/) )

Proof of Theorem en0

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 6885	. . 3 |- ( A ~~ (/) <-> E. f f : A -1-1-onto-> (/) )
2		f1ocnv 5581	. . . . 5 |- ( f : A -1-1-onto-> (/) -> `' f : (/) -1-1-onto-> A )
3		f1o00 5604	. . . . . 6 |- ( `' f : (/) -1-1-onto-> A <-> ( `' f = (/) /\ A = (/) ) )
4	3	simprbi 275	. . . . 5 |- ( `' f : (/) -1-1-onto-> A -> A = (/) )
5	2, 4	syl 14	. . . 4 |- ( f : A -1-1-onto-> (/) -> A = (/) )
6	5	exlimiv 1644	. . 3 |- ( E. f f : A -1-1-onto-> (/) -> A = (/) )
7	1, 6	sylbi 121	. 2 |- ( A ~~ (/) -> A = (/) )
8		0ex 4210	. . . 4 |- (/) e. _V
9	8	enref 6906	. . 3 |- (/) ~~ (/)
10		breq1 4085	. . 3 |- ( A = (/) -> ( A ~~ (/) <-> (/) ~~ (/) ) )
11	9, 10	mpbiri 168	. 2 |- ( A = (/) -> A ~~ (/) )
12	7, 11	impbii 126	1 |- ( A ~~ (/) <-> A = (/) )

Syntax hints:   <-> wb 105   = wceq 1395   E.wex 1538   (/)c0 3491   class class class wbr 4082   `'ccnv 4715   -1-1-onto->wf1o 5313   ~~ cen 6875

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-en 6878

This theorem is referenced by:  nneneq  7006  php5  7007  snnen2oprc  7009  php5dom  7012  ssfilem  7025  dif1enen  7030  fin0  7035  fin0or  7036  diffitest  7037  findcard  7038  findcard2  7039  findcard2s  7040  diffisn  7043  fiintim  7081  fisseneq  7084  fihasheq0  11002  zfz1iso  11050