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Theorem riinn0 4729
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 3956 . 2 (𝐴 𝑥𝑋 𝑆) = ( 𝑥𝑋 𝑆𝐴)
2 r19.2z 4201 . . . . 5 ((𝑋 ≠ ∅ ∧ ∀𝑥𝑋 𝑆𝐴) → ∃𝑥𝑋 𝑆𝐴)
32ancoms 455 . . . 4 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → ∃𝑥𝑋 𝑆𝐴)
4 iinss 4705 . . . 4 (∃𝑥𝑋 𝑆𝐴 𝑥𝑋 𝑆𝐴)
53, 4syl 17 . . 3 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → 𝑥𝑋 𝑆𝐴)
6 df-ss 3737 . . 3 ( 𝑥𝑋 𝑆𝐴 ↔ ( 𝑥𝑋 𝑆𝐴) = 𝑥𝑋 𝑆)
75, 6sylib 208 . 2 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → ( 𝑥𝑋 𝑆𝐴) = 𝑥𝑋 𝑆)
81, 7syl5eq 2817 1 ((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → (𝐴 𝑥𝑋 𝑆) = 𝑥𝑋 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wne 2943  wral 3061  wrex 3062  cin 3722  wss 3723  c0 4063   ciin 4655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-v 3353  df-dif 3726  df-in 3730  df-ss 3737  df-nul 4064  df-iin 4657
This theorem is referenced by:  riinrab  4730  riiner  7972  mreriincl  16466  riinopn  20933  alexsublem  22068  fnemeet1  32698
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