Theorem en0 3328

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 1745	. . . 4 |- (/) e. V
2	1	bren 3282	. . 3 |- (A ~~ (/) <-> E.f f:A-1-1-onto->(/))
3		f1ocnv 2811	. . . . 5 |- (f:A-1-1-onto->(/) -> `'f:(/)-1-1-onto->A)
4		f1o00 2823	. . . . . 6 |- (`'f:(/)-1-1-onto->A <-> (`'f = (/) /\ A = (/)))
5	4	pm3.27bd 263	. . . . 5 |- (`'f:(/)-1-1-onto->A -> A = (/))
6	3, 5	syl 12	. . . 4 |- (f:A-1-1-onto->(/) -> A = (/))
7	6	19.23aiv 952	. . 3 |- (E.f f:A-1-1-onto->(/) -> A = (/))
8	2, 7	sylbi 174	. 2 |- (A ~~ (/) -> A = (/))
9	1	enref 3295	. . 3 |- (/) ~~ (/)
10		breq1 2065	. . 3 |- (A = (/) -> (A ~~ (/) <-> (/) ~~ (/)))
11	9, 10	mpbiri 169	. 2 |- (A = (/) -> A ~~ (/))
12	8, 11	impbi 139	1 |- (A ~~ (/) <-> A = (/))

Syntax hints:   <-> wb 127  E.wex 678   = wceq 1091  (/)c0 1707   class class class wbr 2054  `'ccnv 2409  -1-1-onto->wf1o 2421   ~~ cen 3271

This theorem is referenced by:  snfi 3337  dom0 3367  0sdom 3368  nneneq 3408  fiint 3445  cardeq0 3639  infmap2 4953

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-en 3274

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