Theorem bnj521 30805

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj521	|- ( A i^i { A } ) = (/)

Proof of Theorem bnj521

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elirr 8505	. . . 4 |- -. A e. A
2		elin 3796	. . . . . 6 |- ( x e. ( A i^i { A } ) <-> ( x e. A /\ x e. { A } ) )
3		velsn 4193	. . . . . . 7 |- ( x e. { A } <-> x = A )
4		eleq1 2689	. . . . . . . 8 |- ( x = A -> ( x e. A <-> A e. A ) )
5	4	biimpac 503	. . . . . . 7 |- ( ( x e. A /\ x = A ) -> A e. A )
6	3, 5	sylan2b 492	. . . . . 6 |- ( ( x e. A /\ x e. { A } ) -> A e. A )
7	2, 6	sylbi 207	. . . . 5 |- ( x e. ( A i^i { A } ) -> A e. A )
8	7	exlimiv 1858	. . . 4 |- ( E. x x e. ( A i^i { A } ) -> A e. A )
9	1, 8	mto 188	. . 3 |- -. E. x x e. ( A i^i { A } )
10		n0 3931	. . 3 |- ( ( A i^i { A } ) =/= (/) <-> E. x x e. ( A i^i { A } ) )
11	9, 10	mtbir 313	. 2 |- -. ( A i^i { A } ) =/= (/)
12		nne 2798	. 2 |- ( -. ( A i^i { A } ) =/= (/) <-> ( A i^i { A } ) = (/) )
13	11, 12	mpbi 220	1 |- ( A i^i { A } ) = (/)

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   i^i cin 3573   (/)c0 3915   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by:  bnj927  30839  bnj535  30960