Theorem bj-1uplth 34321

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 34316	. . 3 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → pr1 ⦅𝐴⦆ = pr1 ⦅𝐵⦆)
2		bj-pr11val 34319	. . 3 ⊢ pr1 ⦅𝐴⦆ = 𝐴
3		bj-pr11val 34319	. . 3 ⊢ pr1 ⦅𝐵⦆ = 𝐵
4	1, 2, 3	3eqtr3g 2881	. 2 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → 𝐴 = 𝐵)
5		bj-1upleq 34313	. 2 ⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
6	4, 5	impbii 211	1 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵)

Syntax hints:   ↔ wb 208   = wceq 1537  ⦅bj-c1upl 34311  pr₁ bj-cpr1 34314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-xp 5563  df-rel 5564  df-cnv 5565  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-bj-sngl 34280  df-bj-tag 34289  df-bj-proj 34305  df-bj-1upl 34312  df-bj-pr1 34315

This theorem is referenced by: (None)