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Theorem bj-1uplth 32188
 Description: The characteristic property of monuples. Note that this holds without sethood hypotheses. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-1uplth (⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵)

Proof of Theorem bj-1uplth
StepHypRef Expression
1 bj-pr1eq 32183 . . 3 (⦅𝐴⦆ = ⦅𝐵⦆ → pr1𝐴⦆ = pr1𝐵⦆)
2 bj-pr11val 32186 . . 3 pr1𝐴⦆ = 𝐴
3 bj-pr11val 32186 . . 3 pr1𝐵⦆ = 𝐵
41, 2, 33eqtr3g 2667 . 2 (⦅𝐴⦆ = ⦅𝐵⦆ → 𝐴 = 𝐵)
5 bj-1upleq 32180 . 2 (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
64, 5impbii 198 1 (⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475  ⦅bj-c1upl 32178  pr1 bj-cpr1 32181 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-bj-sngl 32147  df-bj-tag 32156  df-bj-proj 32172  df-bj-1upl 32179  df-bj-pr1 32182 This theorem is referenced by: (None)
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