Theorem npsspw 7654

Description: Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.)

Assertion

Ref	Expression
npsspw	⊢ P ⊆ (𝒫 Q × 𝒫 Q)

Proof of Theorem npsspw

Dummy variables 𝑢 𝑙 𝑞 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 527	. . . 4 ⊢ ((((𝑙 ⊆ Q ∧ 𝑢 ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ 𝑙 ∧ ∃𝑟 ∈ Q 𝑟 ∈ 𝑢)) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)) ∧ ∀𝑟 ∈ Q (𝑟 ∈ 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ 𝑢) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢)))) → (𝑙 ⊆ Q ∧ 𝑢 ⊆ Q))
2		velpw 3656	. . . . 5 ⊢ (𝑙 ∈ 𝒫 Q ↔ 𝑙 ⊆ Q)
3		velpw 3656	. . . . 5 ⊢ (𝑢 ∈ 𝒫 Q ↔ 𝑢 ⊆ Q)
4	2, 3	anbi12i 460	. . . 4 ⊢ ((𝑙 ∈ 𝒫 Q ∧ 𝑢 ∈ 𝒫 Q) ↔ (𝑙 ⊆ Q ∧ 𝑢 ⊆ Q))
5	1, 4	sylibr 134	. . 3 ⊢ ((((𝑙 ⊆ Q ∧ 𝑢 ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ 𝑙 ∧ ∃𝑟 ∈ Q 𝑟 ∈ 𝑢)) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)) ∧ ∀𝑟 ∈ Q (𝑟 ∈ 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ 𝑢) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢)))) → (𝑙 ∈ 𝒫 Q ∧ 𝑢 ∈ 𝒫 Q))
6	5	ssopab2i 4365	. 2 ⊢ {⟨𝑙, 𝑢⟩ ∣ (((𝑙 ⊆ Q ∧ 𝑢 ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ 𝑙 ∧ ∃𝑟 ∈ Q 𝑟 ∈ 𝑢)) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)) ∧ ∀𝑟 ∈ Q (𝑟 ∈ 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ 𝑢) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢))))} ⊆ {⟨𝑙, 𝑢⟩ ∣ (𝑙 ∈ 𝒫 Q ∧ 𝑢 ∈ 𝒫 Q)}
7		df-inp 7649	. 2 ⊢ P = {⟨𝑙, 𝑢⟩ ∣ (((𝑙 ⊆ Q ∧ 𝑢 ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ 𝑙 ∧ ∃𝑟 ∈ Q 𝑟 ∈ 𝑢)) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ 𝑙 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ 𝑙)) ∧ ∀𝑟 ∈ Q (𝑟 ∈ 𝑢 ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ 𝑢))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ 𝑙 ∧ 𝑞 ∈ 𝑢) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ 𝑙 ∨ 𝑟 ∈ 𝑢))))}
8		df-xp 4724	. 2 ⊢ (𝒫 Q × 𝒫 Q) = {⟨𝑙, 𝑢⟩ ∣ (𝑙 ∈ 𝒫 Q ∧ 𝑢 ∈ 𝒫 Q)}
9	6, 7, 8	3sstr4i 3265	1 ⊢ P ⊆ (𝒫 Q × 𝒫 Q)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713   ∧ w3a 1002   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509   ⊆ wss 3197  𝒫 cpw 3649   class class class wbr 4082  {copab 4143   × cxp 4716  Qcnq 7463   <_Q cltq 7468  Pcnp 7474

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-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651  df-opab 4145  df-xp 4724  df-inp 7649

This theorem is referenced by:  preqlu  7655  npex  7656  elinp  7657  prop  7658  elnp1st2nd  7659  cauappcvgprlemladd  7841