Theorem iuneq0 50069

Description: An indexed union is empty iff all indexed classes are empty. (Contributed by Zhi Wang, 1-Nov-2025.)

Assertion

Ref	Expression
iuneq0	⊢ (∀𝑥 ∈ 𝐴 𝐵 = ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 = ∅)

Proof of Theorem iuneq0

Step	Hyp	Ref	Expression
1		iunss 5005	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∅ ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∅)
2		ss0b 4353	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 = ∅)
3		ss0b 4353	. . 3 ⊢ (𝐵 ⊆ ∅ ↔ 𝐵 = ∅)
4	3	ralbii 3111	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ ∅ ↔ ∀𝑥 ∈ 𝐴 𝐵 = ∅)
5	1, 2, 4	3bitr3ri 305	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 = ∅)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569  ∀wral 3079   ⊆ wss 3899  ∅c0 4280  ∪ ciun 4953

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-11 2196  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-v 3455  df-ss 3900  df-dif 4049  df-nul 4281  df-iun 4955

This theorem is used by: (None)