Theorem xpss12 4882

Description: Subset theorem for cross product. Generalization of Theorem 101 of [Suppes] p. 52. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Andrew Salmon, 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 3242	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
2		ssel 3242	. . . 4 ⊢ (𝐶 ⊆ 𝐷 → (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))
3	1, 2	im2anan9 606	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)))
4	3	ssopab2dv 4421	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)})
5		df-xp 4780	. 2 ⊢ (𝐴 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)}
6		df-xp 4780	. 2 ⊢ (𝐵 × 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)}
7	4, 5, 6	3sstr4g 3291	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∈ wcel 2209   ⊆ wss 3220  {copab 4191   × cxp 4772

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  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-nfc 2381  df-in 3226  df-ss 3233  df-opab 4193  df-xp 4780

This theorem is used by:  xpss  4883  xpss1  4885  xpss2  4886  djussxp  4925  ssxpbm  5223  ssrnres  5230  cossxp  5310  cossxp2  5311  cocnvss  5313  relrelss  5314  fssxp  5555  oprabss  6174  pmss12g  6956  caserel  7428  casef  7429  dmaddpi  7693  dmmulpi  7694  rexpssxrxp  8371  ltrelxr  8387  dfz2  9722  phimullem  13025  znleval  15039  txuni2  15409  txbas  15411  neitx  15421  txcnp  15424  cnmpt2res  15450  psmetres2  15486  xmetres2  15532  metres2  15534  xmetresbl  15593  xmettx  15663  qtopbasss  15674  tgqioo  15708  resubmet  15709  limccnp2lem  15829  limccnp2cntop  15830  mpodvdsmulf1o  16206  fsumdvdsmul  16207