Theorem iuneq0 49294

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 4987	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∅ ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∅)
2		ss0b 4341	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 = ∅)
3		ss0b 4341	. . 3 ⊢ (𝐵 ⊆ ∅ ↔ 𝐵 = ∅)
4	3	ralbii 3083	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ ∅ ↔ ∀𝑥 ∈ 𝐴 𝐵 = ∅)
5	1, 2, 4	3bitr3ri 302	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 = ∅)

Syntax hints:   ↔ wb 206   = wceq 1542  ∀wral 3051   ⊆ wss 3889  ∅c0 4273  ∪ ciun 4933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-v 3431  df-dif 3892  df-ss 3906  df-nul 4274  df-iun 4935

This theorem is referenced by: (None)