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Theorem bj-1uplth 35690
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 35685 . . 3 (⦅𝐴⦆ = ⦅𝐵⦆ → pr1𝐴⦆ = pr1𝐵⦆)
2 bj-pr11val 35688 . . 3 pr1𝐴⦆ = 𝐴
3 bj-pr11val 35688 . . 3 pr1𝐵⦆ = 𝐵
41, 2, 33eqtr3g 2794 . 2 (⦅𝐴⦆ = ⦅𝐵⦆ → 𝐴 = 𝐵)
5 bj-1upleq 35682 . 2 (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
64, 5impbii 208 1 (⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  bj-c1upl 35680  pr1 bj-cpr1 35683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-bj-sngl 35649  df-bj-tag 35658  df-bj-proj 35674  df-bj-1upl 35681  df-bj-pr1 35684
This theorem is referenced by: (None)
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