Theorem unidif 2530

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 2529	. . 3 |- (A.x e. A E.y e. (A \ B)x (_ y -> U.A (_ U.(A \ B))
2		difss 2167	. . . 4 |- (A \ B) (_ A
3		uniss 2521	. . . 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 2077	. 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 956  A.wral 1645  E.wrex 1646   \ cdif 2044   (_ wss 2047  U.cuni 2503

This theorem is referenced by:  ordunidif 3005

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-uni 2504

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