Theorem xpsspw 4830

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 4734	. . . 4 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
2		vex 2802	. . . . . . . 8 ⊢ 𝑥 ∈ V
3		vex 2802	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	2, 3	dfop 3855	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
5		snssi 3811	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
6		ssun3 3369	. . . . . . . . . . . . 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 3247	. . . . . . . . . . 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 3673	. . . . . . . . . . . 12 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
12		snssi 3811	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐵 → {𝑦} ⊆ 𝐵)
13		ssun4 3370	. . . . . . . . . . . . . . 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 3378	. . . . . . . . . . . . 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 3270	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ⊆ (𝐴 ∪ 𝐵))
19		sseq1 3247	. . . . . . . . . . 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 722	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}) → 𝑧 ⊆ (𝐴 ∪ 𝐵)))
22		vex 2802	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
23	22	elpr 3687	. . . . . . . . 9 ⊢ (𝑧 ∈ {{𝑥}, {𝑥, 𝑦}} ↔ (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}))
24	22	elpw 3655	. . . . . . . . 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 3230	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴 ∪ 𝐵))
27	4, 26	eqsstrid 3270	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ⊆ 𝒫 (𝐴 ∪ 𝐵))
28		sseq1 3247	. . . . . . 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 1943	. . . 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 3655	. . 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 3228	1 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)

Syntax hints:   ∧ wa 104   ∨ wo 713   = wceq 1395  ∃wex 1538   ∈ wcel 2200   ∪ cun 3195   ⊆ wss 3197  𝒫 cpw 3649  {csn 3666  {cpr 3667  ⟨cop 3669   × cxp 4716

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4145  df-xp 4724

This theorem is referenced by:  unixpss  4831  xpexg  4832