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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 36985 | . . 3 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → pr1 ⦅𝐴⦆ = pr1 ⦅𝐵⦆) | |
2 | bj-pr11val 36988 | . . 3 ⊢ pr1 ⦅𝐴⦆ = 𝐴 | |
3 | bj-pr11val 36988 | . . 3 ⊢ pr1 ⦅𝐵⦆ = 𝐵 | |
4 | 1, 2, 3 | 3eqtr3g 2798 | . 2 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → 𝐴 = 𝐵) |
5 | bj-1upleq 36982 | . 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 1537 ⦅bj-c1upl 36980 pr1 bj-cpr1 36983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-bj-sngl 36949 df-bj-tag 36958 df-bj-proj 36974 df-bj-1upl 36981 df-bj-pr1 36984 |
This theorem is referenced by: (None) |
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