Theorem bj-unexg

Description: Existence of binary unions of sets, proved from ax-bj-bun . (Contributed by BJ, 12-Jan-2025) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-unexg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ∪ 𝐵 ) ∈ V )

Proof

Step	Hyp	Ref	Expression
1		elissetv	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
2		elissetv	⊢ ( 𝐵 ∈ 𝑊 → ∃ 𝑦 𝑦 = 𝐵 )
3		exdistrv	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) )
4		uneq12	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝑥 ∪ 𝑦 ) = ( 𝐴 ∪ 𝐵 ) )
5		ax-bj-bun	⊢ ∀ 𝑥 ∀ 𝑦 ∃ 𝑧 ∀ 𝑡 ( 𝑡 ∈ 𝑧 ↔ ( 𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦 ) )
6	5	spi	⊢ ∀ 𝑦 ∃ 𝑧 ∀ 𝑡 ( 𝑡 ∈ 𝑧 ↔ ( 𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦 ) )
7	6	spi	⊢ ∃ 𝑧 ∀ 𝑡 ( 𝑡 ∈ 𝑧 ↔ ( 𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦 ) )
8		bj-axbun	⊢ ( ( 𝑥 ∪ 𝑦 ) ∈ V ↔ ∃ 𝑧 ∀ 𝑡 ( 𝑡 ∈ 𝑧 ↔ ( 𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦 ) ) )
9	7 8	mpbir	⊢ ( 𝑥 ∪ 𝑦 ) ∈ V
10	4 9	eqeltrrdi	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝐴 ∪ 𝐵 ) ∈ V )
11	10	exlimiv	⊢ ( ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝐴 ∪ 𝐵 ) ∈ V )
12	11	exlimiv	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝐴 ∪ 𝐵 ) ∈ V )
13	3 12	sylbir	⊢ ( ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) → ( 𝐴 ∪ 𝐵 ) ∈ V )
14	1 2 13	syl2an	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ∪ 𝐵 ) ∈ V )