Theorem ssunib 42424

Description: Two ways to say a class is a subclass of a union. (Contributed by RP, 27-Jan-2025.)

Assertion

Ref	Expression
ssunib	⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦)

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

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem ssunib

Step	Hyp	Ref	Expression
1		dfss3 3962	. 2 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵)
2		eluni2 4903	. . 3 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦)
3	2	ralbii 3085	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦)
4	1, 3	bitri 275	1 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦)

Syntax hints:   ↔ wb 205   ∈ wcel 2098  ∀wral 3053  ∃wrex 3062   ⊆ wss 3940  ∪ cuni 4899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-v 3468  df-in 3947  df-ss 3957  df-uni 4900

This theorem is referenced by:  onmaxnelsup  42427  onsupnmax  42432