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Theorem bj-1uplth 34725
 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 34720 . . 3 (⦅𝐴⦆ = ⦅𝐵⦆ → pr1𝐴⦆ = pr1𝐵⦆)
2 bj-pr11val 34723 . . 3 pr1𝐴⦆ = 𝐴
3 bj-pr11val 34723 . . 3 pr1𝐵⦆ = 𝐵
41, 2, 33eqtr3g 2817 . 2 (⦅𝐴⦆ = ⦅𝐵⦆ → 𝐴 = 𝐵)
5 bj-1upleq 34717 . 2 (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
64, 5impbii 212 1 (⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1539  ⦅bj-c1upl 34715  pr1 bj-cpr1 34718 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-opab 5096  df-xp 5531  df-rel 5532  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-bj-sngl 34684  df-bj-tag 34693  df-bj-proj 34709  df-bj-1upl 34716  df-bj-pr1 34719 This theorem is referenced by: (None)
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