Theorem uniss2 4902

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 5009 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 4893	. . . . 5 ⊢ ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ⊆ ∪ 𝐵)
2	1	expcom 418	. . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝑥 ⊆ 𝑦 → 𝑥 ⊆ ∪ 𝐵))
3	2	rexlimiv 3159	. . 3 ⊢ (∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → 𝑥 ⊆ ∪ 𝐵)
4	3	ralimi 3102	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ 𝐵)
5		unissb 4901	. 2 ⊢ (∪ 𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ 𝐵)
6	4, 5	sylibr 237	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∪ 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089   ⊆ wss 3907  ∪ cuni 4867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-v 3459  df-ss 3924  df-uni 4868

This theorem is referenced by:  unidif  4903  coflim  10233  onsssupeqcond  43864