Theorem uniss2 4922

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 5030 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 4913	. . . . 5 ⊢ ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ⊆ ∪ 𝐵)
2	1	expcom 413	. . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝑥 ⊆ 𝑦 → 𝑥 ⊆ ∪ 𝐵))
3	2	rexlimiv 3135	. . 3 ⊢ (∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → 𝑥 ⊆ ∪ 𝐵)
4	3	ralimi 3074	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ 𝐵)
5		unissb 4920	. 2 ⊢ (∪ 𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ 𝐵)
6	4, 5	sylibr 234	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∪ 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061   ⊆ wss 3931  ∪ cuni 4888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-v 3466  df-ss 3948  df-uni 4889

This theorem is referenced by:  unidif  4923  coflim  10280  onsssupeqcond  43271