Theorem iineq0 49618

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 4292	. . . . 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 4961	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
6	5	elv 3460	. . 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 4372	1 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  Vcvv 3455  ∅c0 4286  ∩ ciin 4957

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-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-nul 4287  df-iin 4959

This theorem is referenced by: (None)