Theorem iuneq0 49440

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 5002	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∅ ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∅)
2		ss0b 4355	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 = ∅)
3		ss0b 4355	. . 3 ⊢ (𝐵 ⊆ ∅ ↔ 𝐵 = ∅)
4	3	ralbii 3108	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ ∅ ↔ ∀𝑥 ∈ 𝐴 𝐵 = ∅)
5	1, 2, 4	3bitr3ri 304	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 = ∅)

Syntax hints:   ↔ wb 208   = wceq 1560  ∀wral 3076   ⊆ wss 3904  ∅c0 4285  ∪ ciun 4949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-v 3456  df-dif 3907  df-ss 3921  df-nul 4286  df-iun 4951

This theorem is referenced by: (None)