Theorem unidif 4913

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 4912	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → ∪ 𝐴 ⊆ ∪ (𝐴 ∖ 𝐵))
2		difss 4093	. . . 4 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3	2	unissi 4886	. . 3 ⊢ ∪ (𝐴 ∖ 𝐵) ⊆ ∪ 𝐴
4	1, 3	jctil 529	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → (∪ (𝐴 ∖ 𝐵) ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ ∪ (𝐴 ∖ 𝐵)))
5		eqss 3955	. 2 ⊢ (∪ (𝐴 ∖ 𝐵) = ∪ 𝐴 ↔ (∪ (𝐴 ∖ 𝐵) ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ ∪ (𝐴 ∖ 𝐵)))
6	4, 5	sylibr 237	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → ∪ (𝐴 ∖ 𝐵) = ∪ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∀wral 3082  ∃wrex 3092   ∖ cdif 3905   ⊆ wss 3908  ∪ cuni 4877

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-v 3460  df-dif 3911  df-ss 3925  df-uni 4878

This theorem is used by:  ordunidif  6418