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Theorem riinn0 5083
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 4209 . 2 (𝐴 𝑥𝑋 𝑆) = ( 𝑥𝑋 𝑆𝐴)
2 r19.2z 4495 . . . . 5 ((𝑋 ≠ ∅ ∧ ∀𝑥𝑋 𝑆𝐴) → ∃𝑥𝑋 𝑆𝐴)
32ancoms 458 . . . 4 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → ∃𝑥𝑋 𝑆𝐴)
4 iinss 5056 . . . 4 (∃𝑥𝑋 𝑆𝐴 𝑥𝑋 𝑆𝐴)
53, 4syl 17 . . 3 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → 𝑥𝑋 𝑆𝐴)
6 dfss2 3969 . . 3 ( 𝑥𝑋 𝑆𝐴 ↔ ( 𝑥𝑋 𝑆𝐴) = 𝑥𝑋 𝑆)
75, 6sylib 218 . 2 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → ( 𝑥𝑋 𝑆𝐴) = 𝑥𝑋 𝑆)
81, 7eqtrid 2789 1 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → (𝐴 𝑥𝑋 𝑆) = 𝑥𝑋 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wne 2940  wral 3061  wrex 3070  cin 3950  wss 3951  c0 4333   ciin 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968  df-nul 4334  df-iin 4994
This theorem is referenced by:  riinrab  5084  riiner  8830  mreriincl  17641  riinopn  22914  alexsublem  24052  fnemeet1  36367
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