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Theorem preqlu 6454
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 ((A P B P) → (A = B ↔ ((1stA) = (1stB) (2ndA) = (2ndB))))

Proof of Theorem preqlu
StepHypRef Expression
1 npsspw 6453 . . . . 5 P ⊆ (𝒫 Q × 𝒫 Q)
21sseli 2935 . . . 4 (A PA (𝒫 Q × 𝒫 Q))
3 1st2nd2 5743 . . . 4 (A (𝒫 Q × 𝒫 Q) → A = ⟨(1stA), (2ndA)⟩)
42, 3syl 14 . . 3 (A PA = ⟨(1stA), (2ndA)⟩)
51sseli 2935 . . . 4 (B PB (𝒫 Q × 𝒫 Q))
6 1st2nd2 5743 . . . 4 (B (𝒫 Q × 𝒫 Q) → B = ⟨(1stB), (2ndB)⟩)
75, 6syl 14 . . 3 (B PB = ⟨(1stB), (2ndB)⟩)
84, 7eqeqan12d 2052 . 2 ((A P B P) → (A = B ↔ ⟨(1stA), (2ndA)⟩ = ⟨(1stB), (2ndB)⟩))
9 xp1st 5734 . . . . 5 (A (𝒫 Q × 𝒫 Q) → (1stA) 𝒫 Q)
102, 9syl 14 . . . 4 (A P → (1stA) 𝒫 Q)
11 xp2nd 5735 . . . . 5 (A (𝒫 Q × 𝒫 Q) → (2ndA) 𝒫 Q)
122, 11syl 14 . . . 4 (A P → (2ndA) 𝒫 Q)
13 opthg 3966 . . . 4 (((1stA) 𝒫 Q (2ndA) 𝒫 Q) → (⟨(1stA), (2ndA)⟩ = ⟨(1stB), (2ndB)⟩ ↔ ((1stA) = (1stB) (2ndA) = (2ndB))))
1410, 12, 13syl2anc 391 . . 3 (A P → (⟨(1stA), (2ndA)⟩ = ⟨(1stB), (2ndB)⟩ ↔ ((1stA) = (1stB) (2ndA) = (2ndB))))
1514adantr 261 . 2 ((A P B P) → (⟨(1stA), (2ndA)⟩ = ⟨(1stB), (2ndB)⟩ ↔ ((1stA) = (1stB) (2ndA) = (2ndB))))
168, 15bitrd 177 1 ((A P B P) → (A = B ↔ ((1stA) = (1stB) (2ndA) = (2ndB))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  𝒫 cpw 3351  cop 3370   × cxp 4286  cfv 4845  1st c1st 5707  2nd c2nd 5708  Qcnq 6264  Pcnp 6275
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fv 4853  df-1st 5709  df-2nd 5710  df-inp 6448
This theorem is referenced by:  genpassg  6509  addnqpr1  6541  distrprg  6562  1idpr  6566  ltexpri  6585  addcanprg  6588  recexprlemex  6607  aptipr  6611
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