Theorem xpsspw 4659

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 4563	. . . 4 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
2		vex 2692	. . . . . . . 8 ⊢ 𝑥 ∈ V
3		vex 2692	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	2, 3	dfop 3712	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
5		snssi 3672	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
6		ssun3 3246	. . . . . . . . . . . . 13 ⊢ ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝐵))
7	5, 6	syl 14	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝐵))
8	7	adantr 274	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥} ⊆ (𝐴 ∪ 𝐵))
9		sseq1 3125	. . . . . . . . . . 11 ⊢ (𝑧 = {𝑥} → (𝑧 ⊆ (𝐴 ∪ 𝐵) ↔ {𝑥} ⊆ (𝐴 ∪ 𝐵)))
10	8, 9	syl5ibrcom 156	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = {𝑥} → 𝑧 ⊆ (𝐴 ∪ 𝐵)))
11		df-pr 3539	. . . . . . . . . . . 12 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
12		snssi 3672	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐵 → {𝑦} ⊆ 𝐵)
13		ssun4 3247	. . . . . . . . . . . . . . 15 ⊢ ({𝑦} ⊆ 𝐵 → {𝑦} ⊆ (𝐴 ∪ 𝐵))
14	12, 13	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐵 → {𝑦} ⊆ (𝐴 ∪ 𝐵))
15	7, 14	anim12i 336	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ({𝑥} ⊆ (𝐴 ∪ 𝐵) ∧ {𝑦} ⊆ (𝐴 ∪ 𝐵)))
16		unss 3255	. . . . . . . . . . . . 13 ⊢ (({𝑥} ⊆ (𝐴 ∪ 𝐵) ∧ {𝑦} ⊆ (𝐴 ∪ 𝐵)) ↔ ({𝑥} ∪ {𝑦}) ⊆ (𝐴 ∪ 𝐵))
17	15, 16	sylib 121	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ({𝑥} ∪ {𝑦}) ⊆ (𝐴 ∪ 𝐵))
18	11, 17	eqsstrid 3148	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ⊆ (𝐴 ∪ 𝐵))
19		sseq1 3125	. . . . . . . . . . 11 ⊢ (𝑧 = {𝑥, 𝑦} → (𝑧 ⊆ (𝐴 ∪ 𝐵) ↔ {𝑥, 𝑦} ⊆ (𝐴 ∪ 𝐵)))
20	18, 19	syl5ibrcom 156	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = {𝑥, 𝑦} → 𝑧 ⊆ (𝐴 ∪ 𝐵)))
21	10, 20	jaod 707	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}) → 𝑧 ⊆ (𝐴 ∪ 𝐵)))
22		vex 2692	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
23	22	elpr 3553	. . . . . . . . 9 ⊢ (𝑧 ∈ {{𝑥}, {𝑥, 𝑦}} ↔ (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}))
24	22	elpw 3521	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝒫 (𝐴 ∪ 𝐵) ↔ 𝑧 ⊆ (𝐴 ∪ 𝐵))
25	21, 23, 24	3imtr4g 204	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 ∈ {{𝑥}, {𝑥, 𝑦}} → 𝑧 ∈ 𝒫 (𝐴 ∪ 𝐵)))
26	25	ssrdv 3108	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴 ∪ 𝐵))
27	4, 26	eqsstrid 3148	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ⊆ 𝒫 (𝐴 ∪ 𝐵))
28		sseq1 3125	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ⊆ 𝒫 (𝐴 ∪ 𝐵) ↔ ⟨𝑥, 𝑦⟩ ⊆ 𝒫 (𝐴 ∪ 𝐵)))
29	28	biimpar 295	. . . . . 6 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ⊆ 𝒫 (𝐴 ∪ 𝐵)) → 𝑧 ⊆ 𝒫 (𝐴 ∪ 𝐵))
30	27, 29	sylan2 284	. . . . 5 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑧 ⊆ 𝒫 (𝐴 ∪ 𝐵))
31	30	exlimivv 1869	. . . 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 3521	. . 3 ⊢ (𝑧 ∈ 𝒫 𝒫 (𝐴 ∪ 𝐵) ↔ 𝑧 ⊆ 𝒫 (𝐴 ∪ 𝐵))
34	32, 33	sylibr 133	. 2 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 ∈ 𝒫 𝒫 (𝐴 ∪ 𝐵))
35	34	ssriv 3106	1 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)

Syntax hints:   ∧ wa 103   ∨ wo 698   = wceq 1332  ∃wex 1469   ∈ wcel 1481   ∪ cun 3074   ⊆ wss 3076  𝒫 cpw 3515  {csn 3532  {cpr 3533  ⟨cop 3535   × cxp 4545

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-opab 3998  df-xp 4553

This theorem is referenced by:  unixpss  4660  xpexg  4661