Theorem iineq0 48930

Description: An indexed intersection is empty if one of the intersected classes is empty. (Contributed by Zhi Wang, 30-Oct-2025.)

Assertion

Ref	Expression
iineq0	⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = ∅)

Proof of Theorem iineq0

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nel02 4286	. . . . 5 ⊢ (𝐵 = ∅ → ¬ 𝑦 ∈ 𝐵)
2	1	reximi 3070	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → ∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐵)
3		rexnal 3084	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐵 ↔ ¬ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
4	2, 3	sylib 218	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → ¬ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
5		eliin 4944	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
6	5	elv 3441	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
7	4, 6	sylnibr 329	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵)
8	7	eq0rdv 4354	1 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  Vcvv 3436  ∅c0 4280  ∩ ciin 4940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-v 3438  df-dif 3900  df-nul 4281  df-iin 4942

This theorem is referenced by: (None)