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Theorem preqlu 7244
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 7243 . . . . 5 P ⊆ (𝒫 Q × 𝒫 Q)
21sseli 3061 . . . 4 (𝐴P𝐴 ∈ (𝒫 Q × 𝒫 Q))
3 1st2nd2 6039 . . . 4 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
42, 3syl 14 . . 3 (𝐴P𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
51sseli 3061 . . . 4 (𝐵P𝐵 ∈ (𝒫 Q × 𝒫 Q))
6 1st2nd2 6039 . . . 4 (𝐵 ∈ (𝒫 Q × 𝒫 Q) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
75, 6syl 14 . . 3 (𝐵P𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
84, 7eqeqan12d 2131 . 2 ((𝐴P𝐵P) → (𝐴 = 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩))
9 xp1st 6029 . . . . 5 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (1st𝐴) ∈ 𝒫 Q)
102, 9syl 14 . . . 4 (𝐴P → (1st𝐴) ∈ 𝒫 Q)
11 xp2nd 6030 . . . . 5 (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (2nd𝐴) ∈ 𝒫 Q)
122, 11syl 14 . . . 4 (𝐴P → (2nd𝐴) ∈ 𝒫 Q)
13 opthg 4128 . . . 4 (((1st𝐴) ∈ 𝒫 Q ∧ (2nd𝐴) ∈ 𝒫 Q) → (⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
1410, 12, 13syl2anc 406 . . 3 (𝐴P → (⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
1514adantr 272 . 2 ((𝐴P𝐵P) → (⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
168, 15bitrd 187 1 ((𝐴P𝐵P) → (𝐴 = 𝐵 ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wcel 1463  𝒫 cpw 3478  cop 3498   × cxp 4505  cfv 5091  1st c1st 6002  2nd c2nd 6003  Qcnq 7052  Pcnp 7063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fv 5099  df-1st 6004  df-2nd 6005  df-inp 7238
This theorem is referenced by:  genpassg  7298  addnqpr  7333  mulnqpr  7349  distrprg  7360  1idpr  7364  ltexpri  7385  addcanprg  7388  recexprlemex  7409  aptipr  7413
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