bj-2uplth - Mathbox for BJ

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Theorem bj-2uplth 32202

Description: The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 4871). (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 32183	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr₁ ⦅𝐴, 𝐵⦆ = pr₁ ⦅𝐶, 𝐷⦆)
2		bj-pr21val 32194	. . . 4 ⊢ pr₁ ⦅𝐴, 𝐵⦆ = 𝐴
3		bj-pr21val 32194	. . . 4 ⊢ pr₁ ⦅𝐶, 𝐷⦆ = 𝐶
4	1, 2, 3	3eqtr3g 2667	. . 3 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐴 = 𝐶)
5		bj-pr2eq 32197	. . . 4 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr₂ ⦅𝐴, 𝐵⦆ = pr₂ ⦅𝐶, 𝐷⦆)
6		bj-pr22val 32200	. . . 4 ⊢ pr₂ ⦅𝐴, 𝐵⦆ = 𝐵
7		bj-pr22val 32200	. . . 4 ⊢ pr₂ ⦅𝐶, 𝐷⦆ = 𝐷
8	5, 6, 7	3eqtr3g 2667	. . 3 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐵 = 𝐷)
9	4, 8	jca 553	. 2 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
10		bj-2upleq 32193	. . 3 ⊢ (𝐴 = 𝐶 → (𝐵 = 𝐷 → ⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆))
11	10	imp 444	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆)
12	9, 11	impbii 198	1 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 pr₁ bj-cpr1 32181 ⦅bj-c2uple 32191 pr₂ bj-cpr2 32195

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-suc 5646 df-1o 7447 df-bj-sngl 32147 df-bj-tag 32156 df-bj-proj 32172 df-bj-1upl 32179 df-bj-pr1 32182 df-bj-2upl 32192 df-bj-pr2 32196

This theorem is referenced by: (None)

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