Theorem bj-1uplth 37009

Description: The characteristic property of monuples. Note that this holds without sethood hypotheses. (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-1uplth	⊢ (⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵)

Proof of Theorem bj-1uplth

Step	Hyp	Ref	Expression
1		bj-pr1eq 37004	. . 3 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → pr1 ⦅𝐴⦆ = pr1 ⦅𝐵⦆)
2		bj-pr11val 37007	. . 3 ⊢ pr1 ⦅𝐴⦆ = 𝐴
3		bj-pr11val 37007	. . 3 ⊢ pr1 ⦅𝐵⦆ = 𝐵
4	1, 2, 3	3eqtr3g 2799	. 2 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → 𝐴 = 𝐵)
5		bj-1upleq 37001	. 2 ⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
6	4, 5	impbii 209	1 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵)

Syntax hints:   ↔ wb 206   = wceq 1539  ⦅bj-c1upl 36999  pr₁ bj-cpr1 37002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-bj-sngl 36968  df-bj-tag 36977  df-bj-proj 36993  df-bj-1upl 37000  df-bj-pr1 37003

This theorem is referenced by: (None)