Theorem disj4 4025

Description: Two ways of saying that two classes are disjoint. (Contributed by NM, 21-Mar-2004.)

Assertion

Ref	Expression
disj4	|- ( ( A i^i B ) = (/) <-> -. ( A \ B ) C. A )

Proof of Theorem disj4

Step	Hyp	Ref	Expression
1		disj3 4021	. 2 |- ( ( A i^i B ) = (/) <-> A = ( A \ B ) )
2		eqcom 2629	. 2 |- ( A = ( A \ B ) <-> ( A \ B ) = A )
3		difss 3737	. . . 4 |- ( A \ B ) C_ A
4		dfpss2 3692	. . . 4 |- ( ( A \ B ) C. A <-> ( ( A \ B ) C_ A /\ -. ( A \ B ) = A ) )
5	3, 4	mpbiran 953	. . 3 |- ( ( A \ B ) C. A <-> -. ( A \ B ) = A )
6	5	con2bii 347	. 2 |- ( ( A \ B ) = A <-> -. ( A \ B ) C. A )
7	1, 2, 6	3bitri 286	1 |- ( ( A i^i B ) = (/) <-> -. ( A \ B ) C. A )

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   \ cdif 3571   i^i cin 3573   C_ wss 3574   C. wpss 3575   (/)c0 3915

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

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-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916

This theorem is referenced by:  marypha1lem  8339  infeq5i  8533  wilthlem2  24795  topdifinffinlem  33195