Theorem unidif 2534

Description: If the difference A \ B contains the largest members of A, then the union of the difference is the union of A.

Assertion

Ref	Expression
unidif	|- (A.x e. A E.y e. (A \ B)x (_ 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 2533	. . 3 |- (A.x e. A E.y e. (A \ B)x (_ y -> U.A (_ U.(A \ B))
2		difss 2170	. . . 4 |- (A \ B) (_ A
3		uniss 2525	. . . 4 |- ((A \ B) (_ A -> U.(A \ B) (_ U.A)
4	2, 3	ax-mp 7	. . 3 |- U.(A \ B) (_ U.A
5	1, 4	jctil 292	. 2 |- (A.x e. A E.y e. (A \ B)x (_ y -> (U.(A \ B) (_ U.A /\ U.A (_ U.(A \ B)))
6		eqss 2080	. 2 |- (U.(A \ B) = U.A <-> (U.(A \ B) (_ U.A /\ U.A (_ U.(A \ B)))
7	5, 6	sylibr 200	1 |- (A.x e. A E.y e. (A \ B)x (_ y -> U.(A \ B) = U.A)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958  A.wral 1648  E.wrex 1649   \ cdif 2047   (_ wss 2050  U.cuni 2507

This theorem is referenced by:  ordunidif 3011

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-in 2054  df-ss 2056  df-uni 2508

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