Theorem ssunieq 3925

Description: Relationship implying union. (Contributed by NM, 10-Nov-1999.)

Assertion

Ref	Expression
ssunieq	⊢ ((A ∈ B ∧ ∀x ∈ B x ⊆ A) → A = ∪B)

Distinct variable groups:   x,A   x,B

Proof of Theorem ssunieq

Step	Hyp	Ref	Expression
1		elssuni 3920	. . 3 ⊢ (A ∈ B → A ⊆ ∪B)
2		unissb 3922	. . . 4 ⊢ (∪B ⊆ A ↔ ∀x ∈ B x ⊆ A)
3	2	biimpri 197	. . 3 ⊢ (∀x ∈ B x ⊆ A → ∪B ⊆ A)
4	1, 3	anim12i 549	. 2 ⊢ ((A ∈ B ∧ ∀x ∈ B x ⊆ A) → (A ⊆ ∪B ∧ ∪B ⊆ A))
5		eqss 3288	. 2 ⊢ (A = ∪B ↔ (A ⊆ ∪B ∧ ∪B ⊆ A))
6	4, 5	sylibr 203	1 ⊢ ((A ∈ B ∧ ∀x ∈ B x ⊆ A) → A = ∪B)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615   ⊆ wss 3258  ∪cuni 3892

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 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-uni 3893

This theorem is referenced by:  unimax  3926