Theorem ssunieq 3925

Description: Relationship implying union. (Contributed by NM, 10-Nov-1999.)

Assertion

Ref	Expression
ssunieq	|- ((A e. B /\ A.x e. B x C_ A) -> A = U.B)

Distinct variable groups:   x,A   x,B

Proof of Theorem ssunieq

Step	Hyp	Ref	Expression
1		elssuni 3920	. . 3 |- (A e. B -> A C_ U.B)
2		unissb 3922	. . . 4 |- (U.B C_ A <-> A.x e. B x C_ A)
3	2	biimpri 197	. . 3 |- (A.x e. B x C_ A -> U.B C_ A)
4	1, 3	anim12i 549	. 2 |- ((A e. B /\ A.x e. B x C_ A) -> (A C_ U.B /\ U.B C_ A))
5		eqss 3288	. 2 |- (A = U.B <-> (A C_ U.B /\ U.B C_ A))
6	4, 5	sylibr 203	1 |- ((A e. B /\ A.x e. B x C_ A) -> A = U.B)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710  A.wral 2615   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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-uni 3893

This theorem is referenced by:  unimax  3926