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Theorem preqlu 6724
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 6723 . . . . 5 P ⊆ (𝒫 Q × 𝒫 Q)
21sseli 2996 . . . 4 (𝐴P𝐴 ∈ (𝒫 Q × 𝒫 Q))
3 1st2nd2 5832 . . . 4 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
42, 3syl 14 . . 3 (𝐴P𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
51sseli 2996 . . . 4 (𝐵P𝐵 ∈ (𝒫 Q × 𝒫 Q))
6 1st2nd2 5832 . . . 4 (𝐵 ∈ (𝒫 Q × 𝒫 Q) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
75, 6syl 14 . . 3 (𝐵P𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
84, 7eqeqan12d 2097 . 2 ((𝐴P𝐵P) → (𝐴 = 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩))
9 xp1st 5823 . . . . 5 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (1st𝐴) ∈ 𝒫 Q)
102, 9syl 14 . . . 4 (𝐴P → (1st𝐴) ∈ 𝒫 Q)
11 xp2nd 5824 . . . . 5 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (2nd𝐴) ∈ 𝒫 Q)
122, 11syl 14 . . . 4 (𝐴P → (2nd𝐴) ∈ 𝒫 Q)
13 opthg 4001 . . . 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 1285  wcel 1434  𝒫 cpw 3390  cop 3409   × cxp 4369  cfv 4932  1st c1st 5796  2nd c2nd 5797  Qcnq 6532  Pcnp 6543
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-iota 4897  df-fun 4934  df-fv 4940  df-1st 5798  df-2nd 5799  df-inp 6718
This theorem is referenced by:  genpassg  6778  addnqpr  6813  mulnqpr  6829  distrprg  6840  1idpr  6844  ltexpri  6865  addcanprg  6868  recexprlemex  6889  aptipr  6893
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