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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-1uplth | Structured version Visualization version GIF version | ||
| Description: The characteristic property of monuples. Note that this holds without sethood hypotheses. (Contributed by BJ, 6-Apr-2019.) | 
| Ref | Expression | 
|---|---|
| bj-1uplth | ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵) | 
| 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) | 
| Colors of variables: wff setvar class | 
| 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) | 
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