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Theorem bj-2uplth 37022
Description: The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 5481). (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-2uplth (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem bj-2uplth
StepHypRef Expression
1 bj-pr1eq 37003 . . . 4 (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr1𝐴, 𝐵⦆ = pr1𝐶, 𝐷⦆)
2 bj-pr21val 37014 . . . 4 pr1𝐴, 𝐵⦆ = 𝐴
3 bj-pr21val 37014 . . . 4 pr1𝐶, 𝐷⦆ = 𝐶
41, 2, 33eqtr3g 2800 . . 3 (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐴 = 𝐶)
5 bj-pr2eq 37017 . . . 4 (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr2𝐴, 𝐵⦆ = pr2𝐶, 𝐷⦆)
6 bj-pr22val 37020 . . . 4 pr2𝐴, 𝐵⦆ = 𝐵
7 bj-pr22val 37020 . . . 4 pr2𝐶, 𝐷⦆ = 𝐷
85, 6, 73eqtr3g 2800 . . 3 (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐵 = 𝐷)
94, 8jca 511 . 2 (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → (𝐴 = 𝐶𝐵 = 𝐷))
10 bj-2upleq 37013 . . 3 (𝐴 = 𝐶 → (𝐵 = 𝐷 → ⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆))
1110imp 406 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → ⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆)
129, 11impbii 209 1 (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  pr1 bj-cpr1 37001  bj-c2uple 37011  pr2 bj-cpr2 37015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-1o 8506  df-bj-sngl 36967  df-bj-tag 36976  df-bj-proj 36992  df-bj-1upl 36999  df-bj-pr1 37002  df-bj-2upl 37012  df-bj-pr2 37016
This theorem is referenced by: (None)
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