Theorem bj-2uplth 35988

Description: The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 5476). (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 35969	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr1 ⦅𝐴, 𝐵⦆ = pr1 ⦅𝐶, 𝐷⦆)
2		bj-pr21val 35980	. . . 4 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴
3		bj-pr21val 35980	. . . 4 ⊢ pr1 ⦅𝐶, 𝐷⦆ = 𝐶
4	1, 2, 3	3eqtr3g 2795	. . 3 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐴 = 𝐶)
5		bj-pr2eq 35983	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr2 ⦅𝐴, 𝐵⦆ = pr2 ⦅𝐶, 𝐷⦆)
6		bj-pr22val 35986	. . . 4 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵
7		bj-pr22val 35986	. . . 4 ⊢ pr2 ⦅𝐶, 𝐷⦆ = 𝐷
8	5, 6, 7	3eqtr3g 2795	. . 3 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐵 = 𝐷)
9	4, 8	jca 512	. 2 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
10		bj-2upleq 35979	. . 3 ⊢ (𝐴 = 𝐶 → (𝐵 = 𝐷 → ⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆))
11	10	imp 407	. 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 396   = wceq 1541  pr₁ bj-cpr1 35967  ⦅bj-c2uple 35977  pr₂ bj-cpr2 35981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-suc 6370  df-1o 8468  df-bj-sngl 35933  df-bj-tag 35942  df-bj-proj 35958  df-bj-1upl 35965  df-bj-pr1 35968  df-bj-2upl 35978  df-bj-pr2 35982

This theorem is referenced by: (None)