Theorem uniss2 2533

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 2599 for a generalization to indexed unions.

Assertion

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

Distinct variable groups:   x,A   x,y,B

Proof of Theorem uniss2

Step	Hyp	Ref	Expression
1		ssuni 2526	. . . . 5 ⊢ ((x ⊆ y ⋀ y ∈ B) → x ⊆ ∪B)
2	1	expcom 374	. . . 4 ⊢ (y ∈ B → (x ⊆ y → x ⊆ ∪B))
3	2	r19.23aiv 1746	. . 3 ⊢ (∃y ∈ B x ⊆ y → x ⊆ ∪B)
4	3	r19.20si 1709	. 2 ⊢ (∀x ∈ A ∃y ∈ B x ⊆ y → ∀x ∈ A x ⊆ ∪B)
5		unissb 2532	. 2 ⊢ (∪A ⊆ ∪B ↔ ∀x ∈ A x ⊆ ∪B)
6	4, 5	sylibr 200	1 ⊢ (∀x ∈ A ∃y ∈ B x ⊆ y → ∪A ⊆ ∪B)

Syntax hints:   → wi 3   ∈ wcel 960  ∀wral 1648  ∃wrex 1649   ⊆ wss 2050  ∪cuni 2507

This theorem is referenced by:  unidif 2534

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815  df-in 2054  df-ss 2056  df-uni 2508

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