Theorem relssdv 5797

Description: Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.)

Hypotheses

Ref	Expression
relssdv.1	⊢ (𝜑 → Rel 𝐴)
relssdv.2	⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))

Assertion

Ref	Expression
relssdv	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

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

Proof of Theorem relssdv

Step	Hyp	Ref	Expression
1		relssdv.2	. . 3 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
2	1	alrimivv 1927	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3		relssdv.1	. . 3 ⊢ (𝜑 → Rel 𝐴)
4		ssrel 5791	. . 3 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
5	3, 4	syl 17	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
6	2, 5	mpbird 257	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1537   ∈ wcel 2107   ⊆ wss 3950  ⟨cop 4631  Rel wrel 5689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-ss 3967  df-opab 5205  df-xp 5690  df-rel 5691

This theorem is referenced by:  relssres  6039  poirr2  6143  sofld  6206  relssdmrn  6287  relssdmrnOLD  6288  funcres2  17944  wunfunc  17947  fthres2  17980  pospo  18391  joindmss  18425  meetdmss  18439  clatl  18554  subrgdvds  20587  opsrtoslem2  22081  txcls  23613  txdis1cn  23644  txkgen  23661  qustgplem  24130  metustid  24568  metustexhalf  24570  ovoliunlem1  25538  dvres2  25948  cvmlift2lem12  35320  dib2dim  41246  dih2dimbALTN  41248  dihmeetlem1N  41293  dihglblem5apreN  41294  dihmeetlem13N  41322  dihjatcclem4  41424