Theorem iuneq0 49617

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 5009	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∅ ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∅)
2		ss0b 4358	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 = ∅)
3		ss0b 4358	. . 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 = ∅)

Syntax hints:   ↔ wb 209   = wceq 1570  ∀wral 3079   ⊆ wss 3905  ∅c0 4286  ∪ ciun 4956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-v 3457  df-dif 3908  df-ss 3922  df-nul 4287  df-iun 4958

This theorem is referenced by: (None)