Theorem bj-2uplth 35138

Description: The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 5385). (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-2uplth	⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Proof of Theorem bj-2uplth

Step	Hyp	Ref	Expression
1		bj-pr1eq 35119	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr1 ⦅𝐴, 𝐵⦆ = pr1 ⦅𝐶, 𝐷⦆)
2		bj-pr21val 35130	. . . 4 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴
3		bj-pr21val 35130	. . . 4 ⊢ pr1 ⦅𝐶, 𝐷⦆ = 𝐶
4	1, 2, 3	3eqtr3g 2802	. . 3 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐴 = 𝐶)
5		bj-pr2eq 35133	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr2 ⦅𝐴, 𝐵⦆ = pr2 ⦅𝐶, 𝐷⦆)
6		bj-pr22val 35136	. . . 4 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵
7		bj-pr22val 35136	. . . 4 ⊢ pr2 ⦅𝐶, 𝐷⦆ = 𝐷
8	5, 6, 7	3eqtr3g 2802	. . 3 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐵 = 𝐷)
9	4, 8	jca 511	. 2 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
10		bj-2upleq 35129	. . 3 ⊢ (𝐴 = 𝐶 → (𝐵 = 𝐷 → ⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆))
11	10	imp 406	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆)
12	9, 11	impbii 208	1 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539  pr₁ bj-cpr1 35117  ⦅bj-c2uple 35127  pr₂ bj-cpr2 35131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-suc 6257  df-1o 8267  df-bj-sngl 35083  df-bj-tag 35092  df-bj-proj 35108  df-bj-1upl 35115  df-bj-pr1 35118  df-bj-2upl 35128  df-bj-pr2 35132

This theorem is referenced by: (None)