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Theorem preqlu 6978
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 6977 . . . . 5 P ⊆ (𝒫 Q × 𝒫 Q)
21sseli 3010 . . . 4 (𝐴P𝐴 ∈ (𝒫 Q × 𝒫 Q))
3 1st2nd2 5904 . . . 4 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
42, 3syl 14 . . 3 (𝐴P𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
51sseli 3010 . . . 4 (𝐵P𝐵 ∈ (𝒫 Q × 𝒫 Q))
6 1st2nd2 5904 . . . 4 (𝐵 ∈ (𝒫 Q × 𝒫 Q) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
75, 6syl 14 . . 3 (𝐵P𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
84, 7eqeqan12d 2100 . 2 ((𝐴P𝐵P) → (𝐴 = 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩))
9 xp1st 5895 . . . . 5 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (1st𝐴) ∈ 𝒫 Q)
102, 9syl 14 . . . 4 (𝐴P → (1st𝐴) ∈ 𝒫 Q)
11 xp2nd 5896 . . . . 5 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (2nd𝐴) ∈ 𝒫 Q)
122, 11syl 14 . . . 4 (𝐴P → (2nd𝐴) ∈ 𝒫 Q)
13 opthg 4041 . . . 4 (((1st𝐴) ∈ 𝒫 Q ∧ (2nd𝐴) ∈ 𝒫 Q) → (⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
1410, 12, 13syl2anc 403 . . 3 (𝐴P → (⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
1514adantr 270 . 2 ((𝐴P𝐵P) → (⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
168, 15bitrd 186 1 ((𝐴P𝐵P) → (𝐴 = 𝐵 ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1287  wcel 1436  𝒫 cpw 3415  cop 3434   × cxp 4411  cfv 4983  1st c1st 5868  2nd c2nd 5869  Qcnq 6786  Pcnp 6797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-mpt 3878  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-iota 4948  df-fun 4985  df-fv 4991  df-1st 5870  df-2nd 5871  df-inp 6972
This theorem is referenced by:  genpassg  7032  addnqpr  7067  mulnqpr  7083  distrprg  7094  1idpr  7098  ltexpri  7119  addcanprg  7122  recexprlemex  7143  aptipr  7147
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