Theorem unidif 3881

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 3880	. . 3 |- ( A. x e. A E. y e. ( A \ B ) x C_ y -> U. A C_ U. ( A \ B ) )
2		difss 3298	. . . 4 |- ( A \ B ) C_ A
3	2	unissi 3872	. . 3 |- U. ( A \ B ) C_ U. A
4	1, 3	jctil 312	. 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 3207	. 2 |- ( U. ( A \ B ) = U. A <-> ( U. ( A \ B ) C_ U. A /\ U. A C_ U. ( A \ B ) ) )
6	4, 5	sylibr 134	1 |- ( A. x e. A E. y e. ( A \ B ) x C_ y -> U. ( A \ B ) = U. A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   A.wral 2483   E.wrex 2484   \ cdif 3162   C_ wss 3165   U.cuni 3849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-dif 3167  df-in 3171  df-ss 3178  df-uni 3850

This theorem is referenced by: (None)