Theorem relssi 5737

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	⊢ 𝐴 ⊆ 𝐵

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem relssi

Step	Hyp	Ref	Expression
1		relssi.1	. . 3 ⊢ Rel 𝐴
2		ssrel 5733	. . 3 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
4		relssi.2	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)
5	4	ax-gen 1797	. 2 ⊢ ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)
6	3, 5	mpgbir 1801	1 ⊢ 𝐴 ⊆ 𝐵

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540   ∈ wcel 2114   ⊆ wss 3902  ⟨cop 4587  Rel wrel 5630

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3443  df-ss 3919  df-opab 5162  df-xp 5631  df-rel 5632

This theorem is referenced by:  xpsspw  5759  oprssdm  7541  dftpos4  8189  enssdomOLD  8918  idssen  8938  ttrcltr  9629  txuni2  23513  acycgr0v  35344  prclisacycgr  35347  txpss3v  36072  pprodss4v  36078  bj-idres  37367  xrnss3v  38584  aoprssdm  47515