Theorem elnp1st2nd 7791

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 7786	. . . . 5 ⊢ P ⊆ (𝒫 Q × 𝒫 Q)
2	1	sseli 3234	. . . 4 ⊢ (𝐴 ∈ P → 𝐴 ∈ (𝒫 Q × 𝒫 Q))
3		prop 7790	. . . . . . 7 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
4		elinp 7789	. . . . . . 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 6369	. . . 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 6359	. . . . . . . 8 ⊢ (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (1st ‘𝐴) ∈ 𝒫 Q)
14	13	elpwid 3680	. . . . . . 7 ⊢ (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (1st ‘𝐴) ⊆ Q)
15		xp2nd 6360	. . . . . . . 8 ⊢ (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (2nd ‘𝐴) ∈ 𝒫 Q)
16	15	elpwid 3680	. . . . . . 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 2309	. 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 2203  ∀wral 2520  ∃wrex 2521   ⊆ wss 3211  𝒫 cpw 3669  ⟨cop 3692   class class class wbr 4109   × cxp 4747  ‘cfv 5352  1^st c1st 6332  2^nd c2nd 6333  Qcnq 7595   <_Q cltq 7600  Pcnp 7606

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-1st 6334  df-2nd 6335  df-qs 6773  df-ni 7619  df-nqqs 7663  df-inp 7781

This theorem is referenced by:  addclpr  7852  mulclpr  7887  ltexprlempr  7923  recexprlempr  7947  cauappcvgprlemcl  7968  caucvgprlemcl  7991  caucvgprprlemcl  8019