Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-1uplth Structured version   Visualization version   GIF version

Theorem bj-1uplth 37009
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 37004 . . 3 (⦅𝐴⦆ = ⦅𝐵⦆ → pr1𝐴⦆ = pr1𝐵⦆)
2 bj-pr11val 37007 . . 3 pr1𝐴⦆ = 𝐴
3 bj-pr11val 37007 . . 3 pr1𝐵⦆ = 𝐵
41, 2, 33eqtr3g 2799 . 2 (⦅𝐴⦆ = ⦅𝐵⦆ → 𝐴 = 𝐵)
5 bj-1upleq 37001 . 2 (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
64, 5impbii 209 1 (⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1539  bj-c1upl 36999  pr1 bj-cpr1 37002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-bj-sngl 36968  df-bj-tag 36977  df-bj-proj 36993  df-bj-1upl 37000  df-bj-pr1 37003
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator