Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iin0 Structured version   Visualization version   GIF version

Theorem iin0 5237
 Description: An indexed intersection of the empty set, with a nonempty index set, is empty. (Contributed by NM, 20-Oct-2005.)
Assertion
Ref Expression
iin0 (𝐴 ≠ ∅ ↔ 𝑥𝐴 ∅ = ∅)
Distinct variable group:   𝑥,𝐴

Proof of Theorem iin0
StepHypRef Expression
1 iinconst 4905 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 ∅ = ∅)
2 0ex 5187 . . . . . 6 ∅ ∈ V
32n0ii 4278 . . . . 5 ¬ V = ∅
4 0iin 4963 . . . . . 6 𝑥 ∈ ∅ ∅ = V
54eqeq1i 2825 . . . . 5 ( 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅)
63, 5mtbir 325 . . . 4 ¬ 𝑥 ∈ ∅ ∅ = ∅
7 iineq1 4912 . . . . 5 (𝐴 = ∅ → 𝑥𝐴 ∅ = 𝑥 ∈ ∅ ∅)
87eqeq1d 2822 . . . 4 (𝐴 = ∅ → ( 𝑥𝐴 ∅ = ∅ ↔ 𝑥 ∈ ∅ ∅ = ∅))
96, 8mtbiri 329 . . 3 (𝐴 = ∅ → ¬ 𝑥𝐴 ∅ = ∅)
109necon2ai 3035 . 2 ( 𝑥𝐴 ∅ = ∅ → 𝐴 ≠ ∅)
111, 10impbii 211 1 (𝐴 ≠ ∅ ↔ 𝑥𝐴 ∅ = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   = wceq 1537   ≠ wne 3006  Vcvv 3473  ∅c0 4269  ∩ ciin 4896 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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-nul 5186 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-v 3475  df-dif 3916  df-nul 4270  df-iin 4898 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator