Theorem relss 4862

Description: Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58. (Contributed by NM, 15-Aug-1994.)

Assertion

Ref	Expression
relss	⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴))

Proof of Theorem relss

Step	Hyp	Ref	Expression
1		sstr2 3255	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (V × V) → 𝐴 ⊆ (V × V)))
2		df-rel 4781	. 2 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
3		df-rel 4781	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
4	1, 2, 3	3imtr4g 205	1 ⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴))

This proof depends on syntax axioms:   → wi 4  Vcvv 2821   ⊆ wss 3220   × cxp 4772  Rel wrel 4779

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233  df-rel 4781

This theorem is used by:  relin1  4895  relin2  4896  reldif  4897  relres  5091  iss  5109  cnvdif  5194  funss  5396  funssres  5420  fliftcnv  6001  fliftfun  6002  reltpos  6521  tpostpos  6535  swoer  6835  erinxp  6883  ltrel  8387  lerel  8389  txdis1cn  15379  xmeter  15537  lgsquadlem1  16196  lgsquadlem2  16197