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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 34316 | . . 3 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → pr1 ⦅𝐴⦆ = pr1 ⦅𝐵⦆) | |
2 | bj-pr11val 34319 | . . 3 ⊢ pr1 ⦅𝐴⦆ = 𝐴 | |
3 | bj-pr11val 34319 | . . 3 ⊢ pr1 ⦅𝐵⦆ = 𝐵 | |
4 | 1, 2, 3 | 3eqtr3g 2881 | . 2 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → 𝐴 = 𝐵) |
5 | bj-1upleq 34313 | . 2 ⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆) | |
6 | 4, 5 | impbii 211 | 1 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ⦅bj-c1upl 34311 pr1 bj-cpr1 34314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-bj-sngl 34280 df-bj-tag 34289 df-bj-proj 34305 df-bj-1upl 34312 df-bj-pr1 34315 |
This theorem is referenced by: (None) |
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