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Theorem relssi 5633
 Description: Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)
Hypotheses
Ref Expression
relssi.1 Rel 𝐴
relssi.2 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)
Assertion
Ref Expression
relssi 𝐴𝐵
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem relssi
StepHypRef Expression
1 relssi.1 . . 3 Rel 𝐴
2 ssrel 5630 . . 3 (Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
31, 2ax-mp 5 . 2 (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
4 relssi.2 . . 3 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)
54ax-gen 1797 . 2 𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)
63, 5mpgbir 1801 1 𝐴𝐵
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   ∈ wcel 2115   ⊆ wss 3910  ⟨cop 4546  Rel wrel 5533 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-v 3473  df-in 3917  df-ss 3927  df-opab 5102  df-xp 5534  df-rel 5535 This theorem is referenced by:  xpsspw  5655  oprssdm  7304  dftpos4  7886  enssdom  8509  idssen  8529  txuni2  22148  acycgr0v  32402  prclisacycgr  32405  txpss3v  33346  pprodss4v  33352  bj-idres  34468  xrnss3v  35662  aoprssdm  43549
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