Theorem iineq0 50070

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 4287	. . . . 5 ⊢ (𝐵 = ∅ → ¬ 𝑦 ∈ 𝐵)
2	1	reximi 3103	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → ∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐵)
3		rexnal 3117	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐵 ↔ ¬ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
4	2, 3	sylib 221	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → ¬ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
5		eliin 4958	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
6	5	elv 3458	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
7	4, 6	sylnibr 332	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵)
8	7	eq0rdv 4367	1 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2146  ∀wral 3079  ∃wrex 3089  Vcvv 3453  ∅c0 4280  ∩ ciin 4954

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-v 3455  df-dif 4049  df-nul 4281  df-iin 4956

This theorem is used by: (None)