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Mirrors > Home > ILE Home > Th. List > difprsn2 | GIF version |
Description: Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.) |
Ref | Expression |
---|---|
difprsn2 | ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prcom 3468 | . . 3 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
2 | 1 | difeq1i 3086 | . 2 ⊢ ({𝐴, 𝐵} ∖ {𝐵}) = ({𝐵, 𝐴} ∖ {𝐵}) |
3 | necom 2329 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | |
4 | difprsn1 3525 | . . 3 ⊢ (𝐵 ≠ 𝐴 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴}) | |
5 | 3, 4 | sylbi 119 | . 2 ⊢ (𝐴 ≠ 𝐵 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴}) |
6 | 2, 5 | syl5eq 2125 | 1 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ≠ wne 2245 ∖ cdif 2970 {csn 3398 {cpr 3399 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-sn 3404 df-pr 3405 |
This theorem is referenced by: (None) |
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