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Theorem ssin 3156
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 3123 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
21imbi2i 215 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
32albii 1359 . . 3 (∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ ∀𝑥(𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
4 jcab 535 . . . 4 ((𝑥𝐴 → (𝑥𝐵𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴𝑥𝐶)))
54albii 1359 . . 3 (∀𝑥(𝑥𝐴 → (𝑥𝐵𝑥𝐶)) ↔ ∀𝑥((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴𝑥𝐶)))
6 19.26 1370 . . 3 (∀𝑥((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴𝑥𝐶)) ↔ (∀𝑥(𝑥𝐴𝑥𝐵) ∧ ∀𝑥(𝑥𝐴𝑥𝐶)))
73, 5, 63bitrri 196 . 2 ((∀𝑥(𝑥𝐴𝑥𝐵) ∧ ∀𝑥(𝑥𝐴𝑥𝐶)) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)))
8 dfss2 2931 . . 3 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
9 dfss2 2931 . . 3 (𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶))
108, 9anbi12i 433 . 2 ((𝐴𝐵𝐴𝐶) ↔ (∀𝑥(𝑥𝐴𝑥𝐵) ∧ ∀𝑥(𝑥𝐴𝑥𝐶)))
11 dfss2 2931 . 2 (𝐴 ⊆ (𝐵𝐶) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)))
127, 10, 113bitr4i 201 1 ((𝐴𝐵𝐴𝐶) ↔ 𝐴 ⊆ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wal 1241  wcel 1393  cin 2913  wss 2914
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-in 2921  df-ss 2928
This theorem is referenced by:  ssini  3157  ssind  3158  uneqin  3185  disjpss  3275  trin  3861  pwin  4016  peano5  4308  fin  5063
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