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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 4196 | . . . . 5 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅) | |
2 | disj3 3973 | . . . . 5 ⊢ (({𝐴, 𝐵} ∩ {𝐶}) = ∅ ↔ {𝐴, 𝐵} = ({𝐴, 𝐵} ∖ {𝐶})) | |
3 | 1, 2 | sylib 207 | . . . 4 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → {𝐴, 𝐵} = ({𝐴, 𝐵} ∖ {𝐶})) |
4 | 3 | eqcomd 2616 | . . 3 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} ∖ {𝐶}) = {𝐴, 𝐵}) |
5 | difid 3902 | . . . 4 ⊢ ({𝐶} ∖ {𝐶}) = ∅ | |
6 | 5 | a1i 11 | . . 3 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐶} ∖ {𝐶}) = ∅) |
7 | 4, 6 | uneq12d 3730 | . 2 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → (({𝐴, 𝐵} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶})) = ({𝐴, 𝐵} ∪ ∅)) |
8 | df-tp 4130 | . . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
9 | 8 | difeq1i 3686 | . . 3 ⊢ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = (({𝐴, 𝐵} ∪ {𝐶}) ∖ {𝐶}) |
10 | difundir 3839 | . . 3 ⊢ (({𝐴, 𝐵} ∪ {𝐶}) ∖ {𝐶}) = (({𝐴, 𝐵} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶})) | |
11 | 9, 10 | eqtr2i 2633 | . 2 ⊢ (({𝐴, 𝐵} ∖ {𝐶}) ∪ ({𝐶} ∖ {𝐶})) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) |
12 | un0 3919 | . 2 ⊢ ({𝐴, 𝐵} ∪ ∅) = {𝐴, 𝐵} | |
13 | 7, 11, 12 | 3eqtr3g 2667 | 1 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ≠ wne 2780 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 {csn 4125 {cpr 4127 {ctp 4129 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 df-pr 4128 df-tp 4130 |
This theorem is referenced by: f13dfv 6430 nb3graprlem2 25981 cusgra3v 25993 frgra3v 26529 3vfriswmgra 26532 signswch 29964 signstfvcl 29976 nb3grprlem2 40609 cplgr3v 40657 frgr3v 41445 3vfriswmgr 41448 |
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