Theorem unidif 2534

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 groups:   x,y,A   x,B,y

Proof of Theorem unidif

Step	Hyp	Ref	Expression
1		uniss2 2533	. . 3 ⊢ (∀x ∈ A ∃y ∈ (A ∖ B)x ⊆ y → ∪A ⊆ ∪(A ∖ B))
2		difss 2170	. . . 4 ⊢ (A ∖ B) ⊆ A
3		uniss 2525	. . . 4 ⊢ ((A ∖ B) ⊆ A → ∪(A ∖ B) ⊆ ∪A)
4	2, 3	ax-mp 7	. . 3 ⊢ ∪(A ∖ B) ⊆ ∪A
5	1, 4	jctil 292	. 2 ⊢ (∀x ∈ A ∃y ∈ (A ∖ B)x ⊆ y → (∪(A ∖ B) ⊆ ∪A ⋀ ∪A ⊆ ∪(A ∖ B)))
6		eqss 2080	. 2 ⊢ (∪(A ∖ B) = ∪A ↔ (∪(A ∖ B) ⊆ ∪A ⋀ ∪A ⊆ ∪(A ∖ B)))
7	5, 6	sylibr 200	1 ⊢ (∀x ∈ A ∃y ∈ (A ∖ B)x ⊆ y → ∪(A ∖ B) = ∪A)

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 958  ∀wral 1648  ∃wrex 1649   ∖ cdif 2047   ⊆ wss 2050  ∪cuni 2507

This theorem is referenced by:  ordunidif 3011

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-in 2054  df-ss 2056  df-uni 2508

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