Theorem unidif 3871

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 3870	. . 3 |- ( A. x e. A E. y e. ( A \ B ) x C_ y -> U. A C_ U. ( A \ B ) )
2		difss 3289	. . . 4 |- ( A \ B ) C_ A
3	2	unissi 3862	. . 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 3198	. 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 1364   A.wral 2475   E.wrex 2476   \ cdif 3154   C_ wss 3157   U.cuni 3839

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-uni 3840

This theorem is referenced by: (None)