Theorem iuneq0 50261

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 5008	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∅ ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∅)
2		ss0b 4354	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 = ∅)
3		ss0b 4354	. . 3 ⊢ (𝐵 ⊆ ∅ ↔ 𝐵 = ∅)
4	3	ralbii 3112	. 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 1570  ∀wral 3080   ⊆ wss 3900  ∅c0 4281  ∪ ciun 4956

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-v 3456  df-ss 3901  df-dif 4050  df-nul 4282  df-iun 4958

This theorem is used by: (None)