Theorem disjsn 3432

Description: Intersection with the singleton of a non-member is disjoint. (Contributed by NM, 22-May-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)

Assertion

Ref	Expression
disjsn	|- ( ( A i^i { B } ) = (/) <-> -. B e. A )

Proof of Theorem disjsn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disj1 3270	. 2 |- ( ( A i^i { B } ) = (/) <-> A. x ( x e. A -> -. x e. { B } ) )
2		con2b 593	. . . 4 |- ( ( x e. A -> -. x e. { B } ) <-> ( x e. { B } -> -. x e. A ) )
3		velsn 3392	. . . . 5 |- ( x e. { B } <-> x = B )
4	3	imbi1i 227	. . . 4 |- ( ( x e. { B } -> -. x e. A ) <-> ( x = B -> -. x e. A ) )
5		imnan 624	. . . 4 |- ( ( x = B -> -. x e. A ) <-> -. ( x = B /\ x e. A ) )
6	2, 4, 5	3bitri 195	. . 3 |- ( ( x e. A -> -. x e. { B } ) <-> -. ( x = B /\ x e. A ) )
7	6	albii 1359	. 2 |- ( A. x ( x e. A -> -. x e. { B } ) <-> A. x -. ( x = B /\ x e. A ) )
8		alnex 1388	. . 3 |- ( A. x -. ( x = B /\ x e. A ) <-> -. E. x ( x = B /\ x e. A ) )
9		df-clel 2036	. . 3 |- ( B e. A <-> E. x ( x = B /\ x e. A ) )
10	8, 9	xchbinxr 608	. 2 |- ( A. x -. ( x = B /\ x e. A ) <-> -. B e. A )
11	1, 7, 10	3bitri 195	1 |- ( ( A i^i { B } ) = (/) <-> -. B e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   i^i cin 2916   (/)c0 3224   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-dif 2920  df-in 2924  df-nul 3225  df-sn 3381

This theorem is referenced by:  disjsn2  3433  orddisj  4270  ndmima  4702  funtpg  4950  fnunsn  5006  ressnop0  5344  ftpg  5347  fsnunf  5362  fsnunfv  5363  phpm  6327  fiunsnnn  6338  ac6sfi  6352  fzpreddisj  8933  fzp1disj  8942  frecfzennn  9203