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Mirrors > Home > MPE Home > Th. List > difprsn1 | Structured version Visualization version GIF version |
Description: Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.) |
Ref | Expression |
---|---|
difprsn1 | ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necom 2992 | . 2 ⊢ (𝐵 ≠ 𝐴 ↔ 𝐴 ≠ 𝐵) | |
2 | df-pr 4630 | . . . . . 6 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
3 | 2 | equncomi 4154 | . . . . 5 ⊢ {𝐴, 𝐵} = ({𝐵} ∪ {𝐴}) |
4 | 3 | difeq1i 4117 | . . . 4 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) = (({𝐵} ∪ {𝐴}) ∖ {𝐴}) |
5 | difun2 4479 | . . . 4 ⊢ (({𝐵} ∪ {𝐴}) ∖ {𝐴}) = ({𝐵} ∖ {𝐴}) | |
6 | 4, 5 | eqtri 2758 | . . 3 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) = ({𝐵} ∖ {𝐴}) |
7 | disjsn2 4715 | . . . 4 ⊢ (𝐵 ≠ 𝐴 → ({𝐵} ∩ {𝐴}) = ∅) | |
8 | disj3 4452 | . . . 4 ⊢ (({𝐵} ∩ {𝐴}) = ∅ ↔ {𝐵} = ({𝐵} ∖ {𝐴})) | |
9 | 7, 8 | sylib 217 | . . 3 ⊢ (𝐵 ≠ 𝐴 → {𝐵} = ({𝐵} ∖ {𝐴})) |
10 | 6, 9 | eqtr4id 2789 | . 2 ⊢ (𝐵 ≠ 𝐴 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵}) |
11 | 1, 10 | sylbir 234 | 1 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ≠ wne 2938 ∖ cdif 3944 ∪ cun 3945 ∩ cin 3946 ∅c0 4321 {csn 4627 {cpr 4629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-nul 4322 df-sn 4628 df-pr 4630 |
This theorem is referenced by: difprsn2 4803 f12dfv 7273 pmtrprfval 19396 nbgr2vtx1edg 28874 nbuhgr2vtx1edgb 28876 nfrgr2v 29792 mptprop 32187 cycpm2tr 32548 eulerpartlemgf 33676 coinflippvt 33781 ldepsnlinc 47276 |
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