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Theorem ssin 3153
Description: Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26. (Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
ssin ((AB A𝐶) ↔ A ⊆ (B𝐶))

Proof of Theorem ssin
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elin 3120 . . . . 5 (x (B𝐶) ↔ (x B x 𝐶))
21imbi2i 215 . . . 4 ((x Ax (B𝐶)) ↔ (x A → (x B x 𝐶)))
32albii 1356 . . 3 (x(x Ax (B𝐶)) ↔ x(x A → (x B x 𝐶)))
4 jcab 535 . . . 4 ((x A → (x B x 𝐶)) ↔ ((x Ax B) (x Ax 𝐶)))
54albii 1356 . . 3 (x(x A → (x B x 𝐶)) ↔ x((x Ax B) (x Ax 𝐶)))
6 19.26 1367 . . 3 (x((x Ax B) (x Ax 𝐶)) ↔ (x(x Ax B) x(x Ax 𝐶)))
73, 5, 63bitrri 196 . 2 ((x(x Ax B) x(x Ax 𝐶)) ↔ x(x Ax (B𝐶)))
8 dfss2 2928 . . 3 (ABx(x Ax B))
9 dfss2 2928 . . 3 (A𝐶x(x Ax 𝐶))
108, 9anbi12i 433 . 2 ((AB A𝐶) ↔ (x(x Ax B) x(x Ax 𝐶)))
11 dfss2 2928 . 2 (A ⊆ (B𝐶) ↔ x(x Ax (B𝐶)))
127, 10, 113bitr4i 201 1 ((AB A𝐶) ↔ A ⊆ (B𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   wcel 1390  cin 2910  wss 2911
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-bnd 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-v 2553  df-in 2918  df-ss 2925
This theorem is referenced by:  ssini  3154  ssind  3155  uneqin  3182  disjpss  3272  trin  3855  pwin  4010  peano5  4264  fin  5019
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