Theorem xpss12 4856

Description: Subset theorem for cross product. Generalization of Theorem 101 of [Suppes] p. 52. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by NM, 26-Aug-1995.) (Revised by set.mm contributors, 27-Aug-2011.)

Assertion

Ref	Expression
xpss12	⊢ ((A ⊆ B ∧ C ⊆ D) → (A × C) ⊆ (B × D))

Proof of Theorem xpss12

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3268	. . . 4 ⊢ (A ⊆ B → (x ∈ A → x ∈ B))
2		ssel 3268	. . . 4 ⊢ (C ⊆ D → (y ∈ C → y ∈ D))
3	1, 2	im2anan9 808	. . 3 ⊢ ((A ⊆ B ∧ C ⊆ D) → ((x ∈ A ∧ y ∈ C) → (x ∈ B ∧ y ∈ D)))
4	3	ssopab2dv 4716	. 2 ⊢ ((A ⊆ B ∧ C ⊆ D) → {⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ C)} ⊆ {⟨x, y⟩ ∣ (x ∈ B ∧ y ∈ D)})
5		df-xp 4785	. 2 ⊢ (A × C) = {⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ C)}
6		df-xp 4785	. 2 ⊢ (B × D) = {⟨x, y⟩ ∣ (x ∈ B ∧ y ∈ D)}
7	4, 5, 6	3sstr4g 3313	1 ⊢ ((A ⊆ B ∧ C ⊆ D) → (A × C) ⊆ (B × D))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   ∈ wcel 1710   ⊆ wss 3258  {copab 4623   × cxp 4771

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624  df-xp 4785

This theorem is used by:  xpss1  4857  xpss2  4858  ssxpb  5056  fssxp  5233