Theorem uniss2 3923

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 4012 for a generalization to indexed unions. (Contributed by NM, 22-Mar-2004.)

Assertion

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

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

Allowed substitution hint:   A(y)

Proof of Theorem uniss2

Step	Hyp	Ref	Expression
1		ssuni 3914	. . . . 5 ⊢ ((x ⊆ y ∧ y ∈ B) → x ⊆ ∪B)
2	1	expcom 424	. . . 4 ⊢ (y ∈ B → (x ⊆ y → x ⊆ ∪B))
3	2	rexlimiv 2733	. . 3 ⊢ (∃y ∈ B x ⊆ y → x ⊆ ∪B)
4	3	ralimi 2690	. 2 ⊢ (∀x ∈ A ∃y ∈ B x ⊆ y → ∀x ∈ A x ⊆ ∪B)
5		unissb 3922	. 2 ⊢ (∪A ⊆ ∪B ↔ ∀x ∈ A x ⊆ ∪B)
6	4, 5	sylibr 203	1 ⊢ (∀x ∈ A ∃y ∈ B x ⊆ y → ∪A ⊆ ∪B)

Syntax hints:   → wi 4   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616   ⊆ 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-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-uni 3893

This theorem is referenced by:  unidif  3924