Theorem en0 4413

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 2707	. . . 4 |- (/) e. V
2	1	bren 4368	. . 3 |- (A ~~ (/) <-> E.f f:A-1-1-onto->(/))
3		f1ocnv 3696	. . . . 5 |- (f:A-1-1-onto->(/) -> `'f:(/)-1-1-onto->A)
4		f1o00 3709	. . . . . 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 1294	. . 3 |- (E.f f:A-1-1-onto->(/) -> A = (/))
8	2, 7	sylbi 199	. 2 |- (A ~~ (/) -> A = (/))
9	1	enref 4381	. . 3 |- (/) ~~ (/)
10		breq1 2618	. . 3 |- (A = (/) -> (A ~~ (/) <-> (/) ~~ (/)))
11	9, 10	mpbiri 194	. 2 |- (A = (/) -> A ~~ (/))
12	8, 11	impbi 157	1 |- (A ~~ (/) <-> A = (/))

Syntax hints:   <-> wb 146   = wceq 955  E.wex 979  (/)c0 2277   class class class wbr 2615  `'ccnv 3165  -1-1-onto->wf1o 3177   ~~ cen 4357

This theorem is referenced by:  snfi 4422  dom0 4454  0sdomg 4455  nneneq 4501  unifi 4541  fiint 4543  cardeq0 4815  infmap2 7541

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-en 4360

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