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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 2994 | . 2 ⊢ (𝐵 ≠ 𝐴 ↔ 𝐴 ≠ 𝐵) | |
| 2 | df-pr 4629 | . . . . . 6 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
| 3 | 2 | equncomi 4160 | . . . . 5 ⊢ {𝐴, 𝐵} = ({𝐵} ∪ {𝐴}) |
| 4 | 3 | difeq1i 4122 | . . . 4 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) = (({𝐵} ∪ {𝐴}) ∖ {𝐴}) |
| 5 | difun2 4481 | . . . 4 ⊢ (({𝐵} ∪ {𝐴}) ∖ {𝐴}) = ({𝐵} ∖ {𝐴}) | |
| 6 | 4, 5 | eqtri 2765 | . . 3 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) = ({𝐵} ∖ {𝐴}) |
| 7 | disjsn2 4712 | . . . 4 ⊢ (𝐵 ≠ 𝐴 → ({𝐵} ∩ {𝐴}) = ∅) | |
| 8 | disj3 4454 | . . . 4 ⊢ (({𝐵} ∩ {𝐴}) = ∅ ↔ {𝐵} = ({𝐵} ∖ {𝐴})) | |
| 9 | 7, 8 | sylib 218 | . . 3 ⊢ (𝐵 ≠ 𝐴 → {𝐵} = ({𝐵} ∖ {𝐴})) |
| 10 | 6, 9 | eqtr4id 2796 | . 2 ⊢ (𝐵 ≠ 𝐴 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵}) |
| 11 | 1, 10 | sylbir 235 | 1 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ≠ wne 2940 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 {csn 4626 {cpr 4628 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-nul 4334 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: difprsn2 4801 f12dfv 7293 pmtrprfval 19505 nbgr2vtx1edg 29367 nbuhgr2vtx1edgb 29369 nfrgr2v 30291 mptprop 32707 cycpm2tr 33139 eulerpartlemgf 34381 coinflippvt 34487 ldepsnlinc 48425 |
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