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Theorem riinn0 4040
 Description: Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
riinn0 ((x X S A X) → (Ax X S) = x X S)
Distinct variable groups:   x,A   x,X
Allowed substitution hint:   S(x)

Proof of Theorem riinn0
StepHypRef Expression
1 incom 3448 . 2 (Ax X S) = (x X SA)
2 r19.2z 3639 . . . . 5 ((X x X S A) → x X S A)
32ancoms 439 . . . 4 ((x X S A X) → x X S A)
4 iinss 4017 . . . 4 (x X S Ax X S A)
53, 4syl 15 . . 3 ((x X S A X) → x X S A)
6 df-ss 3259 . . 3 (x X S A ↔ (x X SA) = x X S)
75, 6sylib 188 . 2 ((x X S A X) → (x X SA) = x X S)
81, 7syl5eq 2397 1 ((x X S A X) → (Ax X S) = x X S)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ≠ wne 2516  ∀wral 2614  ∃wrex 2615   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550  ∩ciin 3970 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551  df-iin 3972 This theorem is referenced by:  riinrab  4041
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