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Theorem ssbrd 4061
Description: Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
Hypothesis
Ref Expression
ssbrd.1 (𝜑𝐴𝐵)
Assertion
Ref Expression
ssbrd (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))

Proof of Theorem ssbrd
StepHypRef Expression
1 ssbrd.1 . . 3 (𝜑𝐴𝐵)
21sseld 3169 . 2 (𝜑 → (⟨𝐶, 𝐷⟩ ∈ 𝐴 → ⟨𝐶, 𝐷⟩ ∈ 𝐵))
3 df-br 4019 . 2 (𝐶𝐴𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝐴)
4 df-br 4019 . 2 (𝐶𝐵𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝐵)
52, 3, 43imtr4g 205 1 (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2160  wss 3144  cop 3610   class class class wbr 4018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157  df-br 4019
This theorem is referenced by:  ssbri  4062  sess1  4355  brrelex12  4682  coss1  4800  coss2  4801  eqbrrdva  4815  ersym  6572  ertr  6575  subrguss  13600
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