Theorem relssi 5773

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 5769	. . 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 1823	. 2 ⊢ ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)
6	3, 5	mpgbir 1827	1 ⊢ 𝐴 ⊆ 𝐵

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1566   ∈ wcel 2141   ⊆ wss 3904  ⟨cop 4594  Rel wrel 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-ss 3921  df-opab 5173  df-xp 5667  df-rel 5668

This theorem is referenced by:  xpsspw  5796  oprssdm  7591  dftpos4  8240  enssdomOLD  8973  idssen  8993  ttrcltr  9684  txuni2  23701  acycgr0v  35594  prclisacycgr  35597  txpss3v  36322  pprodss4v  36328  bj-idres  37748  xrnss3v  38976  aoprssdm  47884