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Theorem xpss12 4388
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 ((AB 𝐶𝐷) → (A × 𝐶) ⊆ (B × 𝐷))

Proof of Theorem xpss12
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 2933 . . . 4 (AB → (x Ax B))
2 ssel 2933 . . . 4 (𝐶𝐷 → (y 𝐶y 𝐷))
31, 2im2anan9 530 . . 3 ((AB 𝐶𝐷) → ((x A y 𝐶) → (x B y 𝐷)))
43ssopab2dv 4006 . 2 ((AB 𝐶𝐷) → {⟨x, y⟩ ∣ (x A y 𝐶)} ⊆ {⟨x, y⟩ ∣ (x B y 𝐷)})
5 df-xp 4294 . 2 (A × 𝐶) = {⟨x, y⟩ ∣ (x A y 𝐶)}
6 df-xp 4294 . 2 (B × 𝐷) = {⟨x, y⟩ ∣ (x B y 𝐷)}
74, 5, 63sstr4g 2980 1 ((AB 𝐶𝐷) → (A × 𝐶) ⊆ (B × 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  wss 2911  {copab 3808   × cxp 4286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-opab 3810  df-xp 4294
This theorem is referenced by:  xpss  4389  xpss1  4391  xpss2  4392  djussxp  4424  ssxpbm  4699  ssrnres  4706  cossxp  4786  relrelss  4787  fssxp  5001  oprabss  5532  dmaddpi  6309  dmmulpi  6310  rexpssxrxp  6827  ltrelxr  6837  dfz2  8049
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