Theorem xpss12 4646

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 3091	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
2		ssel 3091	. . . 4 ⊢ (𝐶 ⊆ 𝐷 → (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐷))
3	1, 2	im2anan9 587	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)))
4	3	ssopab2dv 4200	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)})
5		df-xp 4545	. 2 ⊢ (𝐴 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)}
6		df-xp 4545	. 2 ⊢ (𝐵 × 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)}
7	4, 5, 6	3sstr4g 3140	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐷))

Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1480   ⊆ wss 3071  {copab 3988   × cxp 4537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-in 3077  df-ss 3084  df-opab 3990  df-xp 4545

This theorem is referenced by:  xpss  4647  xpss1  4649  xpss2  4650  djussxp  4684  ssxpbm  4974  ssrnres  4981  cossxp  5061  cossxp2  5062  cocnvss  5064  relrelss  5065  fssxp  5290  oprabss  5857  pmss12g  6569  caserel  6972  casef  6973  dmaddpi  7133  dmmulpi  7134  rexpssxrxp  7810  ltrelxr  7825  dfz2  9123  phimullem  11901  txuni2  12425  txbas  12427  neitx  12437  txcnp  12440  cnmpt2res  12466  psmetres2  12502  xmetres2  12548  metres2  12550  xmetresbl  12609  xmettx  12679  qtopbasss  12690  tgqioo  12716  resubmet  12717  limccnp2lem  12814  limccnp2cntop  12815