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Theorem bj-1uplth 36192
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 36187 . . 3 (⦅𝐴⦆ = ⦅𝐵⦆ → pr1𝐴⦆ = pr1𝐵⦆)
2 bj-pr11val 36190 . . 3 pr1𝐴⦆ = 𝐴
3 bj-pr11val 36190 . . 3 pr1𝐵⦆ = 𝐵
41, 2, 33eqtr3g 2794 . 2 (⦅𝐴⦆ = ⦅𝐵⦆ → 𝐴 = 𝐵)
5 bj-1upleq 36184 . 2 (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
64, 5impbii 208 1 (⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1540  bj-c1upl 36182  pr1 bj-cpr1 36185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-bj-sngl 36151  df-bj-tag 36160  df-bj-proj 36176  df-bj-1upl 36183  df-bj-pr1 36186
This theorem is referenced by: (None)
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