Theorem bj-1uplth 36988

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 36983	. . 3 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → pr1 ⦅𝐴⦆ = pr1 ⦅𝐵⦆)
2		bj-pr11val 36986	. . 3 ⊢ pr1 ⦅𝐴⦆ = 𝐴
3		bj-pr11val 36986	. . 3 ⊢ pr1 ⦅𝐵⦆ = 𝐵
4	1, 2, 3	3eqtr3g 2787	. 2 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → 𝐴 = 𝐵)
5		bj-1upleq 36980	. 2 ⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
6	4, 5	impbii 209	1 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵)

Syntax hints:   ↔ wb 206   = wceq 1540  ⦅bj-c1upl 36978  pr₁ bj-cpr1 36981

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 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

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 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-bj-sngl 36947  df-bj-tag 36956  df-bj-proj 36972  df-bj-1upl 36979  df-bj-pr1 36982

This theorem is referenced by: (None)