Theorem uniss2 3827

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. (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 3818	. . . . 5 ⊢ ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ⊆ ∪ 𝐵)
2	1	expcom 115	. . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝑥 ⊆ 𝑦 → 𝑥 ⊆ ∪ 𝐵))
3	2	rexlimiv 2581	. . 3 ⊢ (∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → 𝑥 ⊆ ∪ 𝐵)
4	3	ralimi 2533	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ 𝐵)
5		unissb 3826	. 2 ⊢ (∪ 𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ 𝐵)
6	4, 5	sylibr 133	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∪ 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449   ⊆ wss 3121  ∪ cuni 3796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797

This theorem is referenced by:  unidif  3828