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Theorem ssin 3302
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 ((𝐴𝐵𝐴𝐶) ↔ 𝐴 ⊆ (𝐵𝐶))

Proof of Theorem ssin
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elin 3263 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
21imbi2i 225 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
32albii 1447 . . 3 (∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ ∀𝑥(𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
4 jcab 593 . . . 4 ((𝑥𝐴 → (𝑥𝐵𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴𝑥𝐶)))
54albii 1447 . . 3 (∀𝑥(𝑥𝐴 → (𝑥𝐵𝑥𝐶)) ↔ ∀𝑥((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴𝑥𝐶)))
6 19.26 1458 . . 3 (∀𝑥((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴𝑥𝐶)) ↔ (∀𝑥(𝑥𝐴𝑥𝐵) ∧ ∀𝑥(𝑥𝐴𝑥𝐶)))
73, 5, 63bitrri 206 . 2 ((∀𝑥(𝑥𝐴𝑥𝐵) ∧ ∀𝑥(𝑥𝐴𝑥𝐶)) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)))
8 dfss2 3090 . . 3 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
9 dfss2 3090 . . 3 (𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶))
108, 9anbi12i 456 . 2 ((𝐴𝐵𝐴𝐶) ↔ (∀𝑥(𝑥𝐴𝑥𝐵) ∧ ∀𝑥(𝑥𝐴𝑥𝐶)))
11 dfss2 3090 . 2 (𝐴 ⊆ (𝐵𝐶) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)))
127, 10, 113bitr4i 211 1 ((𝐴𝐵𝐴𝐶) ↔ 𝐴 ⊆ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1330  wcel 1481  cin 3074  wss 3075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3081  df-ss 3088
This theorem is referenced by:  ssini  3303  ssind  3304  uneqin  3331  trin  4043  pwin  4211  peano5  4519  fin  5316  tgval  12255  eltg3i  12262  innei  12369  cnptoprest2  12446
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