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Theorem unidif 3923
 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 (x A y (A B)x y(A B) = A)
Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem unidif
StepHypRef Expression
1 uniss2 3922 . . 3 (x A y (A B)x yA (A B))
2 difss 3393 . . . 4 (A B) A
3 uniss 3912 . . . 4 ((A B) A(A B) A)
42, 3ax-mp 5 . . 3 (A B) A
51, 4jctil 523 . 2 (x A y (A B)x y → ((A B) A A (A B)))
6 eqss 3287 . 2 ((A B) = A ↔ ((A B) A A (A B)))
75, 6sylibr 203 1 (x A y (A B)x y(A B) = A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642  ∀wral 2614  ∃wrex 2615   ∖ cdif 3206   ⊆ wss 3257  ∪cuni 3891 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 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-uni 3892 This theorem is referenced by: (None)
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