Theorem en0 4404

Description: The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88.

Assertion

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

Proof of Theorem en0

Step	Hyp	Ref	Expression
1		0ex 2701	. . . 4 |- (/) e. V
2	1	bren 4359	. . 3 |- (A ~~ (/) <-> E.f f:A-1-1-onto->(/))
3		f1ocnv 3686	. . . . 5 |- (f:A-1-1-onto->(/) -> `'f:(/)-1-1-onto->A)
4		f1o00 3699	. . . . . 6 |- (`'f:(/)-1-1-onto->A <-> (`'f = (/) /\ A = (/)))
5	4	pm3.27bi 326	. . . . 5 |- (`'f:(/)-1-1-onto->A -> A = (/))
6	3, 5	syl 10	. . . 4 |- (f:A-1-1-onto->(/) -> A = (/))
7	6	19.23aiv 1290	. . 3 |- (E.f f:A-1-1-onto->(/) -> A = (/))
8	2, 7	sylbi 199	. 2 |- (A ~~ (/) -> A = (/))
9	1	enref 4372	. . 3 |- (/) ~~ (/)
10		breq1 2612	. . 3 |- (A = (/) -> (A ~~ (/) <-> (/) ~~ (/)))
11	9, 10	mpbiri 194	. 2 |- (A = (/) -> A ~~ (/))
12	8, 11	impbi 157	1 |- (A ~~ (/) <-> A = (/))

Syntax hints:   <-> wb 146   = wceq 953  E.wex 977  (/)c0 2270   class class class wbr 2609  `'ccnv 3159  -1-1-onto->wf1o 3171   ~~ cen 4348

This theorem is referenced by:  snfi 4413  dom0 4445  0sdomg 4446  nneneq 4492  unifi 4532  fiint 4534  cardeq0 4804  infmap2 7523

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-en 4351

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