Theorem unidif 3924

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 3923	. . 3 |- (A.x e. A E.y e. (A \ B)x C_ y -> U.A C_ U.(A \ B))
2		difss 3394	. . . 4 |- (A \ B) C_ A
3		uniss 3913	. . . 4 |- ((A \ B) C_ A -> U.(A \ B) C_ U.A)
4	2, 3	ax-mp 5	. . 3 |- U.(A \ B) C_ U.A
5	1, 4	jctil 523	. 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)))
6		eqss 3288	. 2 |- (U.(A \ B) = U.A <-> (U.(A \ B) C_ U.A /\ U.A C_ U.(A \ B)))
7	5, 6	sylibr 203	1 |- (A.x e. A E.y e. (A \ B)x C_ y -> U.(A \ B) = U.A)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642  A.wral 2615  E.wrex 2616   \ cdif 3207   C_ wss 3258  U.cuni 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-uni 3893

This theorem is referenced by: (None)