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Mirrors > Home > NFE 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 3798 | . . 3 ⊢ {A, B} = {B, A} | |
2 | 1 | difeq1i 3381 | . 2 ⊢ ({A, B} ∖ {B}) = ({B, A} ∖ {B}) |
3 | necom 2597 | . . 3 ⊢ (A ≠ B ↔ B ≠ A) | |
4 | difprsn1 3847 | . . 3 ⊢ (B ≠ A → ({B, A} ∖ {B}) = {A}) | |
5 | 3, 4 | sylbi 187 | . 2 ⊢ (A ≠ B → ({B, A} ∖ {B}) = {A}) |
6 | 2, 5 | syl5eq 2397 | 1 ⊢ (A ≠ B → ({A, B} ∖ {B}) = {A}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ≠ wne 2516 ∖ cdif 3206 {csn 3737 {cpr 3738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 df-pr 3742 |
This theorem is referenced by: (None) |
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