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Theorem riinm 3720
 Description: Relative intersection of an inhabited family. (Contributed by Jim Kingdon, 19-Aug-2018.)
Assertion
Ref Expression
riinm ((x 𝑋 𝑆A x x 𝑋) → (A x 𝑋 𝑆) = x 𝑋 𝑆)
Distinct variable groups:   x,A   x,𝑋
Allowed substitution hint:   𝑆(x)

Proof of Theorem riinm
StepHypRef Expression
1 incom 3123 . 2 (A x 𝑋 𝑆) = ( x 𝑋 𝑆A)
2 r19.2m 3303 . . . . 5 ((x x 𝑋 x 𝑋 𝑆A) → x 𝑋 𝑆A)
32ancoms 255 . . . 4 ((x 𝑋 𝑆A x x 𝑋) → x 𝑋 𝑆A)
4 iinss 3699 . . . 4 (x 𝑋 𝑆A x 𝑋 𝑆A)
53, 4syl 14 . . 3 ((x 𝑋 𝑆A x x 𝑋) → x 𝑋 𝑆A)
6 df-ss 2925 . . 3 ( x 𝑋 𝑆A ↔ ( x 𝑋 𝑆A) = x 𝑋 𝑆)
75, 6sylib 127 . 2 ((x 𝑋 𝑆A x x 𝑋) → ( x 𝑋 𝑆A) = x 𝑋 𝑆)
81, 7syl5eq 2081 1 ((x 𝑋 𝑆A x x 𝑋) → (A x 𝑋 𝑆) = x 𝑋 𝑆)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301   ∩ cin 2910   ⊆ wss 2911  ∩ ciin 3649 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-iin 3651 This theorem is referenced by:  riinerm  6115
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