Theorem unidif 4909

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 4908	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → ∪ 𝐴 ⊆ ∪ (𝐴 ∖ 𝐵))
2		difss 4091	. . . 4 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3	2	unissi 4882	. . 3 ⊢ ∪ (𝐴 ∖ 𝐵) ⊆ ∪ 𝐴
4	1, 3	jctil 528	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → (∪ (𝐴 ∖ 𝐵) ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ ∪ (𝐴 ∖ 𝐵)))
5		eqss 3953	. 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)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  ∀wral 3079  ∃wrex 3089   ∖ cdif 3903   ⊆ wss 3906  ∪ cuni 4873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-v 3457  df-dif 3909  df-ss 3923  df-uni 4874

This theorem is referenced by:  ordunidif  6413