Theorem xpsspw 4775

Description: A cross product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.)

Assertion

Ref	Expression
xpsspw	⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)

Proof of Theorem xpsspw

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxpi 4679	. . . 4 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
2		vex 2766	. . . . . . . 8 ⊢ 𝑥 ∈ V
3		vex 2766	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	2, 3	dfop 3807	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
5		snssi 3766	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
6		ssun3 3328	. . . . . . . . . . . . 13 ⊢ ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝐵))
7	5, 6	syl 14	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝐵))
8	7	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥} ⊆ (𝐴 ∪ 𝐵))
9		sseq1 3206	. . . . . . . . . . 11 ⊢ (𝑧 = {𝑥} → (𝑧 ⊆ (𝐴 ∪ 𝐵) ↔ {𝑥} ⊆ (𝐴 ∪ 𝐵)))
10	8, 9	syl5ibrcom 157	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = {𝑥} → 𝑧 ⊆ (𝐴 ∪ 𝐵)))
11		df-pr 3629	. . . . . . . . . . . 12 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
12		snssi 3766	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐵 → {𝑦} ⊆ 𝐵)
13		ssun4 3329	. . . . . . . . . . . . . . 15 ⊢ ({𝑦} ⊆ 𝐵 → {𝑦} ⊆ (𝐴 ∪ 𝐵))
14	12, 13	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐵 → {𝑦} ⊆ (𝐴 ∪ 𝐵))
15	7, 14	anim12i 338	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ({𝑥} ⊆ (𝐴 ∪ 𝐵) ∧ {𝑦} ⊆ (𝐴 ∪ 𝐵)))
16		unss 3337	. . . . . . . . . . . . 13 ⊢ (({𝑥} ⊆ (𝐴 ∪ 𝐵) ∧ {𝑦} ⊆ (𝐴 ∪ 𝐵)) ↔ ({𝑥} ∪ {𝑦}) ⊆ (𝐴 ∪ 𝐵))
17	15, 16	sylib 122	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ({𝑥} ∪ {𝑦}) ⊆ (𝐴 ∪ 𝐵))
18	11, 17	eqsstrid 3229	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ⊆ (𝐴 ∪ 𝐵))
19		sseq1 3206	. . . . . . . . . . 11 ⊢ (𝑧 = {𝑥, 𝑦} → (𝑧 ⊆ (𝐴 ∪ 𝐵) ↔ {𝑥, 𝑦} ⊆ (𝐴 ∪ 𝐵)))
20	18, 19	syl5ibrcom 157	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = {𝑥, 𝑦} → 𝑧 ⊆ (𝐴 ∪ 𝐵)))
21	10, 20	jaod 718	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}) → 𝑧 ⊆ (𝐴 ∪ 𝐵)))
22		vex 2766	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
23	22	elpr 3643	. . . . . . . . 9 ⊢ (𝑧 ∈ {{𝑥}, {𝑥, 𝑦}} ↔ (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}))
24	22	elpw 3611	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝒫 (𝐴 ∪ 𝐵) ↔ 𝑧 ⊆ (𝐴 ∪ 𝐵))
25	21, 23, 24	3imtr4g 205	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 ∈ {{𝑥}, {𝑥, 𝑦}} → 𝑧 ∈ 𝒫 (𝐴 ∪ 𝐵)))
26	25	ssrdv 3189	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴 ∪ 𝐵))
27	4, 26	eqsstrid 3229	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ⊆ 𝒫 (𝐴 ∪ 𝐵))
28		sseq1 3206	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ⊆ 𝒫 (𝐴 ∪ 𝐵) ↔ ⟨𝑥, 𝑦⟩ ⊆ 𝒫 (𝐴 ∪ 𝐵)))
29	28	biimpar 297	. . . . . 6 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ⊆ 𝒫 (𝐴 ∪ 𝐵)) → 𝑧 ⊆ 𝒫 (𝐴 ∪ 𝐵))
30	27, 29	sylan2 286	. . . . 5 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑧 ⊆ 𝒫 (𝐴 ∪ 𝐵))
31	30	exlimivv 1911	. . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑧 ⊆ 𝒫 (𝐴 ∪ 𝐵))
32	1, 31	syl 14	. . 3 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 ⊆ 𝒫 (𝐴 ∪ 𝐵))
33	22	elpw 3611	. . 3 ⊢ (𝑧 ∈ 𝒫 𝒫 (𝐴 ∪ 𝐵) ↔ 𝑧 ⊆ 𝒫 (𝐴 ∪ 𝐵))
34	32, 33	sylibr 134	. 2 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 ∈ 𝒫 𝒫 (𝐴 ∪ 𝐵))
35	34	ssriv 3187	1 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)

Syntax hints:   ∧ wa 104   ∨ wo 709   = wceq 1364  ∃wex 1506   ∈ wcel 2167   ∪ cun 3155   ⊆ wss 3157  𝒫 cpw 3605  {csn 3622  {cpr 3623  ⟨cop 3625   × cxp 4661

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-opab 4095  df-xp 4669

This theorem is referenced by:  unixpss  4776  xpexg  4777