Theorem unidif 1943

Description: If the difference A ∖ B contains the largest members of A, then the union of the difference is the union of A.

Assertion

Ref	Expression
unidif	⊢ (∀x ∈ A ∃y ∈ (A ∖ B)x ⊆ y → ∪(A ∖ B) = ∪A)

Distinct variable group(s):   x,y,A   x,B,y

Proof of Theorem unidif

Step	Hyp	Ref	Expression
1		uniss2 1942	. . 3 ⊢ (∀x ∈ A ∃y ∈ (A ∖ B)x ⊆ y → ∪A ⊆ ∪(A ∖ B))
2		difss 1596	. . . 4 ⊢ (A ∖ B) ⊆ A
3		uniss 1936	. . . 4 ⊢ ((A ∖ B) ⊆ A → ∪(A ∖ B) ⊆ ∪A)
4	2, 3	ax-mp 6	. . 3 ⊢ ∪(A ∖ B) ⊆ ∪A
5	1, 4	jctil 240	. 2 ⊢ (∀x ∈ A ∃y ∈ (A ∖ B)x ⊆ y → (∪(A ∖ B) ⊆ ∪A ∧ ∪A ⊆ ∪(A ∖ B)))
6		eqss 1516	. 2 ⊢ (∪(A ∖ B) = ∪A ↔ (∪(A ∖ B) ⊆ ∪A ∧ ∪A ⊆ ∪(A ∖ B)))
7	5, 6	sylibr 175	1 ⊢ (∀x ∈ A ∃y ∈ (A ∖ B)x ⊆ y → ∪(A ∖ B) = ∪A)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091  ∀wral 1201  ∃wrex 1202   ∖ cdif 1484   ⊆ wss 1487  ∪cuni 1919

This theorem is referenced by:  ordunidif 2260

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-uni 1920

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