Theorem ssunieq 1945

Description: Relationship implying union.

Assertion

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

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

Proof of Theorem ssunieq

Step	Hyp	Ref	Expression
1		elssuni 1940	. . 3 ⊢ (A ∈ B → A ⊆ ∪B)
2		unissb 1941	. . . 4 ⊢ (∪B ⊆ A ↔ ∀x ∈ B x ⊆ A)
3	2	biimpr 134	. . 3 ⊢ (∀x ∈ B x ⊆ A → ∪B ⊆ A)
4	1, 3	anim12i 268	. 2 ⊢ ((A ∈ B ∧ ∀x ∈ B x ⊆ A) → (A ⊆ ∪B ∧ ∪B ⊆ A))
5		eqss 1516	. 2 ⊢ (A = ∪B ↔ (A ⊆ ∪B ∧ ∪B ⊆ A))
6	4, 5	sylibr 175	1 ⊢ ((A ∈ B ∧ ∀x ∈ B x ⊆ A) → A = ∪B)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201   ⊆ wss 1487  ∪cuni 1919

This theorem is referenced by:  unisseq 1946  shsspwh 5153

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-v 1349  df-in 1491  df-ss 1492  df-uni 1920

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