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Mirrors > Home > MPE Home > Th. List > difsnid | Structured version Visualization version 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 3719 | . 2 ⊢ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = ({𝐵} ∪ (𝐴 ∖ {𝐵})) | |
2 | snssi 4280 | . . 3 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴) | |
3 | undif 4001 | . . 3 ⊢ ({𝐵} ⊆ 𝐴 ↔ ({𝐵} ∪ (𝐴 ∖ {𝐵})) = 𝐴) | |
4 | 2, 3 | sylib 207 | . 2 ⊢ (𝐵 ∈ 𝐴 → ({𝐵} ∪ (𝐴 ∖ {𝐵})) = 𝐴) |
5 | 1, 4 | syl5eq 2656 | 1 ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 |
This theorem is referenced by: fnsnsplit 6355 fsnunf2 6357 difsnexi 6864 difsnen 7927 enfixsn 7954 phplem2 8025 pssnn 8063 dif1en 8078 frfi 8090 xpfi 8116 dif1card 8716 hashfun 13084 fprodfvdvdsd 14896 prmdvdsprmo 15584 mreexexlem4d 16130 symgextf1 17664 symgextfo 17665 symgfixf1 17680 gsumdifsnd 18183 gsummgp0 18431 islindf4 19996 scmatf1 20156 gsummatr01 20284 tdeglem4 23624 dfconngra1 26199 lindsenlbs 32574 poimirlem25 32604 poimirlem27 32606 hdmap14lem4a 36181 hdmap14lem13 36190 fsumnncl 38638 hoidmv1lelem2 39482 dfconngr1 41355 mgpsumunsn 41933 gsumdifsndf 41937 |
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