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Theorem preqlu 6568
Description: Two reals are equal if and only if their lower and upper cuts are. (Contributed by Jim Kingdon, 11-Dec-2019.)
Assertion
Ref Expression
preqlu ((𝐴P𝐵P) → (𝐴 = 𝐵 ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))

Proof of Theorem preqlu
StepHypRef Expression
1 npsspw 6567 . . . . 5 P ⊆ (𝒫 Q × 𝒫 Q)
21sseli 2941 . . . 4 (𝐴P𝐴 ∈ (𝒫 Q × 𝒫 Q))
3 1st2nd2 5801 . . . 4 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
42, 3syl 14 . . 3 (𝐴P𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
51sseli 2941 . . . 4 (𝐵P𝐵 ∈ (𝒫 Q × 𝒫 Q))
6 1st2nd2 5801 . . . 4 (𝐵 ∈ (𝒫 Q × 𝒫 Q) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
75, 6syl 14 . . 3 (𝐵P𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
84, 7eqeqan12d 2055 . 2 ((𝐴P𝐵P) → (𝐴 = 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩))
9 xp1st 5792 . . . . 5 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (1st𝐴) ∈ 𝒫 Q)
102, 9syl 14 . . . 4 (𝐴P → (1st𝐴) ∈ 𝒫 Q)
11 xp2nd 5793 . . . . 5 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (2nd𝐴) ∈ 𝒫 Q)
122, 11syl 14 . . . 4 (𝐴P → (2nd𝐴) ∈ 𝒫 Q)
13 opthg 3975 . . . 4 (((1st𝐴) ∈ 𝒫 Q ∧ (2nd𝐴) ∈ 𝒫 Q) → (⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
1410, 12, 13syl2anc 391 . . 3 (𝐴P → (⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
1514adantr 261 . 2 ((𝐴P𝐵P) → (⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
168, 15bitrd 177 1 ((𝐴P𝐵P) → (𝐴 = 𝐵 ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wcel 1393  𝒫 cpw 3359  cop 3378   × cxp 4343  cfv 4902  1st c1st 5765  2nd c2nd 5766  Qcnq 6376  Pcnp 6387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-1st 5767  df-2nd 5768  df-inp 6562
This theorem is referenced by:  genpassg  6622  addnqpr  6657  mulnqpr  6673  distrprg  6684  1idpr  6688  ltexpri  6709  addcanprg  6712  recexprlemex  6733  aptipr  6737
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