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

Step	Hyp	Ref	Expression
1		uniss2 3922	. . 3 ⊢ (∀x ∈ A ∃y ∈ (A ∖ B)x ⊆ y → ∪A ⊆ ∪(A ∖ B))
2		difss 3393	. . . 4 ⊢ (A ∖ B) ⊆ A
3		uniss 3912	. . . 4 ⊢ ((A ∖ B) ⊆ A → ∪(A ∖ B) ⊆ ∪A)
4	2, 3	ax-mp 5	. . 3 ⊢ ∪(A ∖ B) ⊆ ∪A
5	1, 4	jctil 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)))
7	5, 6	sylibr 203	1 ⊢ (∀x ∈ A ∃y ∈ (A ∖ B)x ⊆ y → ∪(A ∖ B) = ∪A)

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)