Theorem iin0imm 4211

Description: An indexed intersection of the empty set, with an inhabited index set, is empty. (Contributed by Jim Kingdon, 29-Aug-2018.)

Assertion

Ref	Expression
iin0imm	⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 ∅ = ∅)

Distinct variable groups:   𝑦,𝐴   𝑥,𝐴

Proof of Theorem iin0imm

Step	Hyp	Ref	Expression
1		iinconstm 3935	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 ∅ = ∅)

Syntax hints:   → wi 4   = wceq 1372  ∃wex 1514   ∈ wcel 2175  ∅c0 3459  ∩ ciin 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-iin 3929

This theorem is referenced by: (None)