Theorem bj-1uplth 37591

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 37586	. . 3 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → pr1 ⦅𝐴⦆ = pr1 ⦅𝐵⦆)
2		bj-pr11val 37589	. . 3 ⊢ pr1 ⦅𝐴⦆ = 𝐴
3		bj-pr11val 37589	. . 3 ⊢ pr1 ⦅𝐵⦆ = 𝐵
4	1, 2, 3	3eqtr3g 2828	. 2 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → 𝐴 = 𝐵)
5		bj-1upleq 37583	. 2 ⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
6	4, 5	impbii 212	1 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵)

Syntax hints:   ↔ wb 209   = wceq 1568  ⦅bj-c1upl 37581  pr₁ bj-cpr1 37584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5671  df-rel 5672  df-cnv 5673  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-bj-sngl 37550  df-bj-tag 37559  df-bj-proj 37575  df-bj-1upl 37582  df-bj-pr1 37585

This theorem is referenced by: (None)