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Theorem riinn0 4910
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 4105 . 2 (𝐴 𝑥𝑋 𝑆) = ( 𝑥𝑋 𝑆𝐴)
2 r19.2z 4360 . . . . 5 ((𝑋 ≠ ∅ ∧ ∀𝑥𝑋 𝑆𝐴) → ∃𝑥𝑋 𝑆𝐴)
32ancoms 459 . . . 4 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → ∃𝑥𝑋 𝑆𝐴)
4 iinss 4885 . . . 4 (∃𝑥𝑋 𝑆𝐴 𝑥𝑋 𝑆𝐴)
53, 4syl 17 . . 3 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → 𝑥𝑋 𝑆𝐴)
6 df-ss 3880 . . 3 ( 𝑥𝑋 𝑆𝐴 ↔ ( 𝑥𝑋 𝑆𝐴) = 𝑥𝑋 𝑆)
75, 6sylib 219 . 2 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → ( 𝑥𝑋 𝑆𝐴) = 𝑥𝑋 𝑆)
81, 7syl5eq 2845 1 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → (𝐴 𝑥𝑋 𝑆) = 𝑥𝑋 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1525  wne 2986  wral 3107  wrex 3108  cin 3864  wss 3865  c0 4217   ciin 4832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-v 3442  df-dif 3868  df-in 3872  df-ss 3880  df-nul 4218  df-iin 4834
This theorem is referenced by:  riinrab  4911  riiner  8227  mreriincl  16702  riinopn  21204  alexsublem  22340  fnemeet1  33325
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