Theorem unidif 4234

Description: If the difference A \ B contains the largest members of A, then the union of the difference is the union of A. (Contributed by NM, 22-Mar-2004.)

Assertion

Ref	Expression
unidif	|- ( A. x e. A E. y e. ( A \ B ) x C_ y -> U. ( A \ B ) = U. A )

Distinct variable groups:   x, y, A   x, B, y

Proof of Theorem unidif

Step	Hyp	Ref	Expression
1		uniss2 4233	. . 3 |- ( A. x e. A E. y e. ( A \ B ) x C_ y -> U. A C_ U. ( A \ B ) )
2		difss 3562	. . . 4 |- ( A \ B ) C_ A
3	2	unissi 4224	. . 3 |- U. ( A \ B ) C_ U. A
4	1, 3	jctil 540	. 2 |- ( A. x e. A E. y e. ( A \ B ) x C_ y -> ( U. ( A \ B ) C_ U. A /\ U. A C_ U. ( A \ B ) ) )
5		eqss 3449	. 2 |- ( U. ( A \ B ) = U. A <-> ( U. ( A \ B ) C_ U. A /\ U. A C_ U. ( A \ B ) ) )
6	4, 5	sylibr 216	1 |- ( A. x e. A E. y e. ( A \ B ) x C_ y -> U. ( A \ B ) = U. A )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1446   A.wral 2739   E.wrex 2740   \ cdif 3403   C_ wss 3406   U.cuni 4201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ral 2744  df-rex 2745  df-v 3049  df-dif 3409  df-in 3413  df-ss 3420  df-uni 4202

This theorem is referenced by:  ordunidif  5474