MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  riinn0 Structured version   Visualization version   GIF version

Theorem riinn0 5088
Description: Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
riinn0 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → (𝐴 𝑥𝑋 𝑆) = 𝑥𝑋 𝑆)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋
Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem riinn0
StepHypRef Expression
1 incom 4217 . 2 (𝐴 𝑥𝑋 𝑆) = ( 𝑥𝑋 𝑆𝐴)
2 r19.2z 4501 . . . . 5 ((𝑋 ≠ ∅ ∧ ∀𝑥𝑋 𝑆𝐴) → ∃𝑥𝑋 𝑆𝐴)
32ancoms 458 . . . 4 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → ∃𝑥𝑋 𝑆𝐴)
4 iinss 5061 . . . 4 (∃𝑥𝑋 𝑆𝐴 𝑥𝑋 𝑆𝐴)
53, 4syl 17 . . 3 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → 𝑥𝑋 𝑆𝐴)
6 dfss2 3981 . . 3 ( 𝑥𝑋 𝑆𝐴 ↔ ( 𝑥𝑋 𝑆𝐴) = 𝑥𝑋 𝑆)
75, 6sylib 218 . 2 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → ( 𝑥𝑋 𝑆𝐴) = 𝑥𝑋 𝑆)
81, 7eqtrid 2787 1 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → (𝐴 𝑥𝑋 𝑆) = 𝑥𝑋 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wne 2938  wral 3059  wrex 3068  cin 3962  wss 3963  c0 4339   ciin 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-iin 4999
This theorem is referenced by:  riinrab  5089  riiner  8829  mreriincl  17643  riinopn  22930  alexsublem  24068  fnemeet1  36349
  Copyright terms: Public domain W3C validator