Theorem bj-1uplth 37210

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 37205	. . 3 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → pr1 ⦅𝐴⦆ = pr1 ⦅𝐵⦆)
2		bj-pr11val 37208	. . 3 ⊢ pr1 ⦅𝐴⦆ = 𝐴
3		bj-pr11val 37208	. . 3 ⊢ pr1 ⦅𝐵⦆ = 𝐵
4	1, 2, 3	3eqtr3g 2795	. 2 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → 𝐴 = 𝐵)
5		bj-1upleq 37202	. 2 ⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
6	4, 5	impbii 209	1 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵)

Syntax hints:   ↔ wb 206   = wceq 1542  ⦅bj-c1upl 37200  pr₁ bj-cpr1 37203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-bj-sngl 37169  df-bj-tag 37178  df-bj-proj 37194  df-bj-1upl 37201  df-bj-pr1 37204

This theorem is referenced by: (None)