bj-2uplth - Mathbox for BJ

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Theorem bj-2uplth 33009

Description: The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 4945). (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 32990	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr₁ ⦅𝐴, 𝐵⦆ = pr₁ ⦅𝐶, 𝐷⦆)
2		bj-pr21val 33001	. . . 4 ⊢ pr₁ ⦅𝐴, 𝐵⦆ = 𝐴
3		bj-pr21val 33001	. . . 4 ⊢ pr₁ ⦅𝐶, 𝐷⦆ = 𝐶
4	1, 2, 3	3eqtr3g 2679	. . 3 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐴 = 𝐶)
5		bj-pr2eq 33004	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr₂ ⦅𝐴, 𝐵⦆ = pr₂ ⦅𝐶, 𝐷⦆)
6		bj-pr22val 33007	. . . 4 ⊢ pr₂ ⦅𝐴, 𝐵⦆ = 𝐵
7		bj-pr22val 33007	. . . 4 ⊢ pr₂ ⦅𝐶, 𝐷⦆ = 𝐷
8	5, 6, 7	3eqtr3g 2679	. . 3 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐵 = 𝐷)
9	4, 8	jca 554	. 2 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
10		bj-2upleq 33000	. . 3 ⊢ (𝐴 = 𝐶 → (𝐵 = 𝐷 → ⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆))
11	10	imp 445	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆)
12	9, 11	impbii 199	1 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 pr₁ bj-cpr1 32988 ⦅bj-c2uple 32998 pr₂ bj-cpr2 33002

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-suc 5729 df-1o 7560 df-bj-sngl 32954 df-bj-tag 32963 df-bj-proj 32979 df-bj-1upl 32986 df-bj-pr1 32989 df-bj-2upl 32999 df-bj-pr2 33003

This theorem is referenced by: (None)

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