Theorem xpss12 4880

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 4419	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)})
5		df-xp 4778	. 2 ⊢ (𝐴 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)}
6		df-xp 4778	. 2 ⊢ (𝐵 × 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)}
7	4, 5, 6	3sstr4g 3291	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2209   ⊆ wss 3220  {copab 4189   × cxp 4770

This theorem was proved from 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 theorem 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 4191  df-xp 4778

This theorem is referenced by:  xpss  4881  xpss1  4883  xpss2  4884  djussxp  4923  ssxpbm  5221  ssrnres  5228  cossxp  5308  cossxp2  5309  cocnvss  5311  relrelss  5312  fssxp  5553  oprabss  6168  pmss12g  6950  caserel  7421  casef  7422  dmaddpi  7686  dmmulpi  7687  rexpssxrxp  8364  ltrelxr  8380  dfz2  9700  phimullem  12986  znleval  14971  txuni2  15340  txbas  15342  neitx  15352  txcnp  15355  cnmpt2res  15381  psmetres2  15417  xmetres2  15463  metres2  15465  xmetresbl  15524  xmettx  15594  qtopbasss  15605  tgqioo  15639  resubmet  15640  limccnp2lem  15760  limccnp2cntop  15761  mpodvdsmulf1o  16087  fsumdvdsmul  16088