Theorem bj-2uplth 37044

Description: The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 5456). (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 37025	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr1 ⦅𝐴, 𝐵⦆ = pr1 ⦅𝐶, 𝐷⦆)
2		bj-pr21val 37036	. . . 4 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴
3		bj-pr21val 37036	. . . 4 ⊢ pr1 ⦅𝐶, 𝐷⦆ = 𝐶
4	1, 2, 3	3eqtr3g 2794	. . 3 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐴 = 𝐶)
5		bj-pr2eq 37039	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr2 ⦅𝐴, 𝐵⦆ = pr2 ⦅𝐶, 𝐷⦆)
6		bj-pr22val 37042	. . . 4 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵
7		bj-pr22val 37042	. . . 4 ⊢ pr2 ⦅𝐶, 𝐷⦆ = 𝐷
8	5, 6, 7	3eqtr3g 2794	. . 3 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐵 = 𝐷)
9	4, 8	jca 511	. 2 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
10		bj-2upleq 37035	. . 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 209	1 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  pr₁ bj-cpr1 37023  ⦅bj-c2uple 37033  pr₂ bj-cpr2 37037

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-suc 6363  df-1o 8485  df-bj-sngl 36989  df-bj-tag 36998  df-bj-proj 37014  df-bj-1upl 37021  df-bj-pr1 37024  df-bj-2upl 37034  df-bj-pr2 37038

This theorem is referenced by: (None)