Theorem relssi 5743

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 5739	. . 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 3889  ⟨cop 4573  Rel wrel 5636

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-ss 3906  df-opab 5148  df-xp 5637  df-rel 5638

This theorem is referenced by:  xpsspw  5765  oprssdm  7548  dftpos4  8195  enssdomOLD  8924  idssen  8944  ttrcltr  9637  txuni2  23530  acycgr0v  35330  prclisacycgr  35333  txpss3v  36058  pprodss4v  36064  bj-idres  37474  xrnss3v  38702  aoprssdm  47650