Theorem ssxp 2487

Description: Subset theorem for cross product. Generalization of Theorem 101 of [Suppes] p. 52.

Assertion

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

Proof of Theorem ssxp

Step	Hyp	Ref	Expression
1		relxp 2486	. . 3 ⊢ Rel (A × C)
2	1	a1i 7	. 2 ⊢ ((A ⊆ B ∧ C ⊆ D) → Rel (A × C))
3		prth 429	. . . 4 ⊢ (((x ∈ A → x ∈ B) ∧ (y ∈ C → y ∈ D)) → ((x ∈ A ∧ y ∈ C) → (x ∈ B ∧ y ∈ D)))
4		visset 1350	. . . . 5 ⊢ y ∈ V
5	4	opelxp 2452	. . . 4 ⊢ (⟨x, y⟩ ∈ (A × C) ↔ (x ∈ A ∧ y ∈ C))
6	4	opelxp 2452	. . . 4 ⊢ (⟨x, y⟩ ∈ (B × D) ↔ (x ∈ B ∧ y ∈ D))
7	3, 5, 6	3imtr4g 426	. . 3 ⊢ (((x ∈ A → x ∈ B) ∧ (y ∈ C → y ∈ D)) → (⟨x, y⟩ ∈ (A × C) → ⟨x, y⟩ ∈ (B × D)))
8		ssel 1502	. . 3 ⊢ (A ⊆ B → (x ∈ A → x ∈ B))
9		ssel 1502	. . 3 ⊢ (C ⊆ D → (y ∈ C → y ∈ D))
10	7, 8, 9	syl2an 349	. 2 ⊢ ((A ⊆ B ∧ C ⊆ D) → (⟨x, y⟩ ∈ (A × C) → ⟨x, y⟩ ∈ (B × D)))
11	2, 10	relssdv 2482	1 ⊢ ((A ⊆ B ∧ C ⊆ D) → (A × C) ⊆ (B × D))

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092   ⊆ wss 1487  ⟨cop 1810   × cxp 2408  Rel wrel 2415

This theorem is referenced by:  ssres2 2590  resabs2 2593  coexg 2671  fssxp 2761  xpdom3 3347  dmaddpi 3812  dmmulpi 3813  axresscn 4062  xpnnen 4927  infxpidmlem7 4939

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424  df-rel 2425

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