Theorem uniss2 1942

Description: A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. See iunss2 2021 for a generalization to indexed unions.

Assertion

Ref	Expression
uniss2	⊢ (∀x ∈ A ∃y ∈ B x ⊆ y → ∪A ⊆ ∪B)

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

Proof of Theorem uniss2

Step	Hyp	Ref	Expression
1		ssuni 1937	. . . . . 6 ⊢ ((x ⊆ y ∧ y ∈ B) → x ⊆ ∪B)
2	1	exp 291	. . . . 5 ⊢ (x ⊆ y → (y ∈ B → x ⊆ ∪B))
3	2	com12 13	. . . 4 ⊢ (y ∈ B → (x ⊆ y → x ⊆ ∪B))
4	3	r19.23aiv 1284	. . 3 ⊢ (∃y ∈ B x ⊆ y → x ⊆ ∪B)
5	4	r19.20si 1254	. 2 ⊢ (∀x ∈ A ∃y ∈ B x ⊆ y → ∀x ∈ A x ⊆ ∪B)
6		unissb 1941	. 2 ⊢ (∪A ⊆ ∪B ↔ ∀x ∈ A x ⊆ ∪B)
7	5, 6	sylibr 175	1 ⊢ (∀x ∈ A ∃y ∈ B x ⊆ y → ∪A ⊆ ∪B)

Syntax hints:   → wi 2   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ⊆ wss 1487  ∪cuni 1919

This theorem is referenced by:  unidif 1943

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-rex 1206  df-v 1349  df-in 1491  df-ss 1492  df-uni 1920

metamath.org