Theorem ssunib 43834

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 3934	. 2 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵)
2		eluni2 4877	. . 3 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦)
3	2	ralbii 3117	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦)
4	1, 3	bitri 278	1 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦)

Syntax hints:   ↔ wb 209   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095   ⊆ wss 3913  ∪ cuni 4873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-v 3465  df-ss 3930  df-uni 4874

This theorem is referenced by:  onmaxnelsup  43837  onsupnmax  43842