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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 32183 | . . 3 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → pr1 ⦅𝐴⦆ = pr1 ⦅𝐵⦆) | |
2 | bj-pr11val 32186 | . . 3 ⊢ pr1 ⦅𝐴⦆ = 𝐴 | |
3 | bj-pr11val 32186 | . . 3 ⊢ pr1 ⦅𝐵⦆ = 𝐵 | |
4 | 1, 2, 3 | 3eqtr3g 2667 | . 2 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → 𝐴 = 𝐵) |
5 | bj-1upleq 32180 | . 2 ⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆) | |
6 | 4, 5 | impbii 198 | 1 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ⦅bj-c1upl 32178 pr1 bj-cpr1 32181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-bj-sngl 32147 df-bj-tag 32156 df-bj-proj 32172 df-bj-1upl 32179 df-bj-pr1 32182 |
This theorem is referenced by: (None) |
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