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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 35685 | . . 3 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → pr1 ⦅𝐴⦆ = pr1 ⦅𝐵⦆) | |
2 | bj-pr11val 35688 | . . 3 ⊢ pr1 ⦅𝐴⦆ = 𝐴 | |
3 | bj-pr11val 35688 | . . 3 ⊢ pr1 ⦅𝐵⦆ = 𝐵 | |
4 | 1, 2, 3 | 3eqtr3g 2794 | . 2 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → 𝐴 = 𝐵) |
5 | bj-1upleq 35682 | . 2 ⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆) | |
6 | 4, 5 | impbii 208 | 1 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ⦅bj-c1upl 35680 pr1 bj-cpr1 35683 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-bj-sngl 35649 df-bj-tag 35658 df-bj-proj 35674 df-bj-1upl 35681 df-bj-pr1 35684 |
This theorem is referenced by: (None) |
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