Theorem elnp1st2nd 7739

Description: Membership in positive reals, using 1^st and 2^nd to refer to the lower and upper cut. (Contributed by Jim Kingdon, 3-Oct-2019.)

Assertion

Ref	Expression
elnp1st2nd	⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐴))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐴))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐴) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐴)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐴) ∨ 𝑟 ∈ (2nd ‘𝐴))))))

Distinct variable group:   𝑟,𝑞,𝐴

Proof of Theorem elnp1st2nd

Step	Hyp	Ref	Expression
1		npsspw 7734	. . . . 5 ⊢ P ⊆ (𝒫 Q × 𝒫 Q)
2	1	sseli 3224	. . . 4 ⊢ (𝐴 ∈ P → 𝐴 ∈ (𝒫 Q × 𝒫 Q))
3		prop 7738	. . . . . . 7 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
4		elinp 7737	. . . . . . 7 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ↔ ((((1st ‘𝐴) ⊆ Q ∧ (2nd ‘𝐴) ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐴))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐴))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐴) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐴)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐴) ∨ 𝑟 ∈ (2nd ‘𝐴))))))
5	3, 4	sylib 122	. . . . . 6 ⊢ (𝐴 ∈ P → ((((1st ‘𝐴) ⊆ Q ∧ (2nd ‘𝐴) ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐴))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐴))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐴) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐴)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐴) ∨ 𝑟 ∈ (2nd ‘𝐴))))))
6	5	simpld 112	. . . . 5 ⊢ (𝐴 ∈ P → (((1st ‘𝐴) ⊆ Q ∧ (2nd ‘𝐴) ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐴))))
7	6	simprd 114	. . . 4 ⊢ (𝐴 ∈ P → (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐴)))
8	2, 7	jca 306	. . 3 ⊢ (𝐴 ∈ P → (𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐴))))
9	5	simprd 114	. . 3 ⊢ (𝐴 ∈ P → ((∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐴))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐴) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐴)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐴) ∨ 𝑟 ∈ (2nd ‘𝐴)))))
10	8, 9	jca 306	. 2 ⊢ (𝐴 ∈ P → ((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐴))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐴))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐴) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐴)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐴) ∨ 𝑟 ∈ (2nd ‘𝐴))))))
11		1st2nd2 6347	. . . 4 ⊢ (𝐴 ∈ (𝒫 Q × 𝒫 Q) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
12	11	ad2antrr 488	. . 3 ⊢ (((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐴))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐴))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐴) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐴)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐴) ∨ 𝑟 ∈ (2nd ‘𝐴))))) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
13		xp1st 6337	. . . . . . . 8 ⊢ (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (1st ‘𝐴) ∈ 𝒫 Q)
14	13	elpwid 3667	. . . . . . 7 ⊢ (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (1st ‘𝐴) ⊆ Q)
15		xp2nd 6338	. . . . . . . 8 ⊢ (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (2nd ‘𝐴) ∈ 𝒫 Q)
16	15	elpwid 3667	. . . . . . 7 ⊢ (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (2nd ‘𝐴) ⊆ Q)
17	14, 16	jca 306	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 Q × 𝒫 Q) → ((1st ‘𝐴) ⊆ Q ∧ (2nd ‘𝐴) ⊆ Q))
18	17	anim1i 340	. . . . 5 ⊢ ((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐴))) → (((1st ‘𝐴) ⊆ Q ∧ (2nd ‘𝐴) ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐴))))
19	18	anim1i 340	. . . 4 ⊢ (((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐴))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐴))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐴) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐴)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐴) ∨ 𝑟 ∈ (2nd ‘𝐴))))) → ((((1st ‘𝐴) ⊆ Q ∧ (2nd ‘𝐴) ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐴))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐴))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐴) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐴)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐴) ∨ 𝑟 ∈ (2nd ‘𝐴))))))
20	19, 4	sylibr 134	. . 3 ⊢ (((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐴))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐴))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐴) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐴)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐴) ∨ 𝑟 ∈ (2nd ‘𝐴))))) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
21	12, 20	eqeltrd 2308	. 2 ⊢ (((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐴))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐴))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐴) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐴)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐴) ∨ 𝑟 ∈ (2nd ‘𝐴))))) → 𝐴 ∈ P)
22	10, 21	impbii 126	1 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐴))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐴))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐴) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐴)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐴)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐴) ∨ 𝑟 ∈ (2nd ‘𝐴))))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512   ⊆ wss 3201  𝒫 cpw 3656  ⟨cop 3676   class class class wbr 4093   × cxp 4729  ‘cfv 5333  1^st c1st 6310  2^nd c2nd 6311  Qcnq 7543   <_Q cltq 7548  Pcnp 7554

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1st 6312  df-2nd 6313  df-qs 6751  df-ni 7567  df-nqqs 7611  df-inp 7729

This theorem is referenced by:  addclpr  7800  mulclpr  7835  ltexprlempr  7871  recexprlempr  7895  cauappcvgprlemcl  7916  caucvgprlemcl  7939  caucvgprprlemcl  7967