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Mirrors > Home > NFE Home > Th. List > difprsn1 | 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 2597 | . 2 ⊢ (B ≠ A ↔ A ≠ B) | |
2 | disjsn2 3787 | . . . 4 ⊢ (B ≠ A → ({B} ∩ {A}) = ∅) | |
3 | disj3 3595 | . . . 4 ⊢ (({B} ∩ {A}) = ∅ ↔ {B} = ({B} ∖ {A})) | |
4 | 2, 3 | sylib 188 | . . 3 ⊢ (B ≠ A → {B} = ({B} ∖ {A})) |
5 | df-pr 3742 | . . . . . 6 ⊢ {A, B} = ({A} ∪ {B}) | |
6 | 5 | equncomi 3410 | . . . . 5 ⊢ {A, B} = ({B} ∪ {A}) |
7 | 6 | difeq1i 3381 | . . . 4 ⊢ ({A, B} ∖ {A}) = (({B} ∪ {A}) ∖ {A}) |
8 | difun2 3629 | . . . 4 ⊢ (({B} ∪ {A}) ∖ {A}) = ({B} ∖ {A}) | |
9 | 7, 8 | eqtri 2373 | . . 3 ⊢ ({A, B} ∖ {A}) = ({B} ∖ {A}) |
10 | 4, 9 | syl6reqr 2404 | . 2 ⊢ (B ≠ A → ({A, B} ∖ {A}) = {B}) |
11 | 1, 10 | sylbir 204 | 1 ⊢ (A ≠ B → ({A, B} ∖ {A}) = {B}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ≠ wne 2516 ∖ cdif 3206 ∪ cun 3207 ∩ cin 3208 ∅c0 3550 {csn 3737 {cpr 3738 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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: difprsn2 3848 |
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