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Mirrors > Home > MPE Home > Th. List > diftpsn3 | Structured version Visualization version GIF version |
Description: Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.) (Proof shortened by JJ, 23-Jul-2021.) |
Ref | Expression |
---|---|
diftpsn3 | ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjprsn 4557 | . . . . 5 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅) | |
2 | disj3 4317 | . . . . 5 ⊢ (({𝐴, 𝐵} ∩ {𝐶}) = ∅ ↔ {𝐴, 𝐵} = ({𝐴, 𝐵} ∖ {𝐶})) | |
3 | 1, 2 | sylib 219 | . . . 4 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → {𝐴, 𝐵} = ({𝐴, 𝐵} ∖ {𝐶})) |
4 | 3 | eqcomd 2801 | . . 3 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} ∖ {𝐶}) = {𝐴, 𝐵}) |
5 | difid 4250 | . . . 4 ⊢ ({𝐶} ∖ {𝐶}) = ∅ | |
6 | 5 | a1i 11 | . . 3 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐶} ∖ {𝐶}) = ∅) |
7 | 4, 6 | uneq12d 4061 | . 2 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → (({𝐴, 𝐵} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶})) = ({𝐴, 𝐵} ∪ ∅)) |
8 | df-tp 4477 | . . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
9 | 8 | difeq1i 4016 | . . 3 ⊢ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = (({𝐴, 𝐵} ∪ {𝐶}) ∖ {𝐶}) |
10 | difundir 4177 | . . 3 ⊢ (({𝐴, 𝐵} ∪ {𝐶}) ∖ {𝐶}) = (({𝐴, 𝐵} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶})) | |
11 | 9, 10 | eqtr2i 2820 | . 2 ⊢ (({𝐴, 𝐵} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶})) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) |
12 | un0 4264 | . 2 ⊢ ({𝐴, 𝐵} ∪ ∅) = {𝐴, 𝐵} | |
13 | 7, 11, 12 | 3eqtr3g 2854 | 1 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ≠ wne 2984 ∖ cdif 3856 ∪ cun 3857 ∩ cin 3858 ∅c0 4211 {csn 4472 {cpr 4474 {ctp 4476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rab 3114 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-sn 4473 df-pr 4475 df-tp 4477 |
This theorem is referenced by: f13dfv 6896 nb3grprlem2 26846 cplgr3v 26900 frgr3v 27746 3vfriswmgr 27749 signswch 31448 signstfvcl 31460 |
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