Theorem unissint 4501

Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4514). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
unissint	|- ( U. A C_ |^| A <-> ( A = (/) \/ U. A = |^| A ) )

Proof of Theorem unissint

Step	Hyp	Ref	Expression
1		simpl 473	. . . . 5 |- ( ( U. A C_ |^| A /\ -. A = (/) ) -> U. A C_ |^| A )
2		df-ne 2795	. . . . . . 7 |- ( A =/= (/) <-> -. A = (/) )
3		intssuni 4499	. . . . . . 7 |- ( A =/= (/) -> |^| A C_ U. A )
4	2, 3	sylbir 225	. . . . . 6 |- ( -. A = (/) -> |^| A C_ U. A )
5	4	adantl 482	. . . . 5 |- ( ( U. A C_ |^| A /\ -. A = (/) ) -> |^| A C_ U. A )
6	1, 5	eqssd 3620	. . . 4 |- ( ( U. A C_ |^| A /\ -. A = (/) ) -> U. A = |^| A )
7	6	ex 450	. . 3 |- ( U. A C_ |^| A -> ( -. A = (/) -> U. A = |^| A ) )
8	7	orrd 393	. 2 |- ( U. A C_ |^| A -> ( A = (/) \/ U. A = |^| A ) )
9		ssv 3625	. . . . 5 |- U. A C_ _V
10		int0 4490	. . . . 5 |- |^| (/) = _V
11	9, 10	sseqtr4i 3638	. . . 4 |- U. A C_ |^| (/)
12		inteq 4478	. . . 4 |- ( A = (/) -> |^| A = |^| (/) )
13	11, 12	syl5sseqr 3654	. . 3 |- ( A = (/) -> U. A C_ |^| A )
14		eqimss 3657	. . 3 |- ( U. A = |^| A -> U. A C_ |^| A )
15	13, 14	jaoi 394	. 2 |- ( ( A = (/) \/ U. A = |^| A ) -> U. A C_ |^| A )
16	8, 15	impbii 199	1 |- ( U. A C_ |^| A <-> ( A = (/) \/ U. A = |^| A ) )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   =/= wne 2794   _Vcvv 3200   C_ wss 3574   (/)c0 3915   U.cuni 4436   |^|cint 4475

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-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-uni 4437  df-int 4476

This theorem is referenced by: (None)