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Theorem bj-2uplth 32656
Description: The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 4905). (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 32637 . . . 4 (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr1𝐴, 𝐵⦆ = pr1𝐶, 𝐷⦆)
2 bj-pr21val 32648 . . . 4 pr1𝐴, 𝐵⦆ = 𝐴
3 bj-pr21val 32648 . . . 4 pr1𝐶, 𝐷⦆ = 𝐶
41, 2, 33eqtr3g 2678 . . 3 (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐴 = 𝐶)
5 bj-pr2eq 32651 . . . 4 (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr2𝐴, 𝐵⦆ = pr2𝐶, 𝐷⦆)
6 bj-pr22val 32654 . . . 4 pr2𝐴, 𝐵⦆ = 𝐵
7 bj-pr22val 32654 . . . 4 pr2𝐶, 𝐷⦆ = 𝐷
85, 6, 73eqtr3g 2678 . . 3 (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐵 = 𝐷)
94, 8jca 554 . 2 (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → (𝐴 = 𝐶𝐵 = 𝐷))
10 bj-2upleq 32647 . . 3 (𝐴 = 𝐶 → (𝐵 = 𝐷 → ⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆))
1110imp 445 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → ⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆)
129, 11impbii 199 1 (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  pr1 bj-cpr1 32635  bj-c2uple 32645  pr2 bj-cpr2 32649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-suc 5688  df-1o 7505  df-bj-sngl 32601  df-bj-tag 32610  df-bj-proj 32626  df-bj-1upl 32633  df-bj-pr1 32636  df-bj-2upl 32646  df-bj-pr2 32650
This theorem is referenced by: (None)
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