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Mirrors > Home > NFE Home > Th. List > difsnid | GIF version |
Description: If we remove a single element from a class then put it back in, we end up with the original class. (Contributed by NM, 2-Oct-2006.) |
Ref | Expression |
---|---|
difsnid | ⊢ (B ∈ A → ((A ∖ {B}) ∪ {B}) = A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 3408 | . 2 ⊢ ((A ∖ {B}) ∪ {B}) = ({B} ∪ (A ∖ {B})) | |
2 | snssi 3852 | . . 3 ⊢ (B ∈ A → {B} ⊆ A) | |
3 | undif 3630 | . . 3 ⊢ ({B} ⊆ A ↔ ({B} ∪ (A ∖ {B})) = A) | |
4 | 2, 3 | sylib 188 | . 2 ⊢ (B ∈ A → ({B} ∪ (A ∖ {B})) = A) |
5 | 1, 4 | syl5eq 2397 | 1 ⊢ (B ∈ A → ((A ∖ {B}) ∪ {B}) = A) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ∈ wcel 1710 ∖ cdif 3206 ∪ cun 3207 ⊆ wss 3257 {csn 3737 |
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-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 |
This theorem is referenced by: pwadjoin 4119 phiall 4618 |
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