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Theorem bj-2uplth 34337
Description: The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 5371). (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 34318 . . . 4 (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr1𝐴, 𝐵⦆ = pr1𝐶, 𝐷⦆)
2 bj-pr21val 34329 . . . 4 pr1𝐴, 𝐵⦆ = 𝐴
3 bj-pr21val 34329 . . . 4 pr1𝐶, 𝐷⦆ = 𝐶
41, 2, 33eqtr3g 2882 . . 3 (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐴 = 𝐶)
5 bj-pr2eq 34332 . . . 4 (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr2𝐴, 𝐵⦆ = pr2𝐶, 𝐷⦆)
6 bj-pr22val 34335 . . . 4 pr2𝐴, 𝐵⦆ = 𝐵
7 bj-pr22val 34335 . . . 4 pr2𝐶, 𝐷⦆ = 𝐷
85, 6, 73eqtr3g 2882 . . 3 (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐵 = 𝐷)
94, 8jca 514 . 2 (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → (𝐴 = 𝐶𝐵 = 𝐷))
10 bj-2upleq 34328 . . 3 (𝐴 = 𝐶 → (𝐵 = 𝐷 → ⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆))
1110imp 409 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → ⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆)
129, 11impbii 211 1 (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1536  pr1 bj-cpr1 34316  bj-c2uple 34326  pr2 bj-cpr2 34330
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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333  ax-un 7464
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-tr 5176  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-ord 6197  df-on 6198  df-suc 6200  df-1o 8105  df-bj-sngl 34282  df-bj-tag 34291  df-bj-proj 34307  df-bj-1upl 34314  df-bj-pr1 34317  df-bj-2upl 34327  df-bj-pr2 34331
This theorem is referenced by: (None)
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