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Theorem ssrin 3159
Description: Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
ssrin (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Proof of Theorem ssrin
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssel 2936 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21anim1d 319 . . 3 (𝐴𝐵 → ((𝑥𝐴𝑥𝐶) → (𝑥𝐵𝑥𝐶)))
3 elin 3123 . . 3 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
4 elin 3123 . . 3 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
52, 3, 43imtr4g 194 . 2 (𝐴𝐵 → (𝑥 ∈ (𝐴𝐶) → 𝑥 ∈ (𝐵𝐶)))
65ssrdv 2948 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  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:  sslin  3160  ss2in  3161  ssdisj  3274  ssdifin0  3301  ssres  4615  phplem2  6294
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