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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 | snssi 4812 | . 2 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴) | |
2 | undifr 4483 | . 2 ⊢ ({𝐵} ⊆ 𝐴 ↔ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴) | |
3 | 1, 2 | sylib 217 | 1 ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∖ cdif 3946 ∪ cun 3947 ⊆ wss 3949 {csn 4629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-sn 4630 |
This theorem is referenced by: fnsnsplit 7182 fsnunf2 7184 difsnexi 7748 difsnen 9053 enfixsn 9081 pssnn 9168 phplem2OLD 9218 pssnnOLD 9265 dif1ennnALT 9277 frfi 9288 xpfiOLD 9318 dif1card 10005 hashgt23el 14384 hashfun 14397 fprodfvdvdsd 16277 prmdvdsprmo 16975 mreexexlem4d 17591 symgextf1 19289 symgextfo 19290 symgfixf1 19305 gsumdifsnd 19829 gsummgp0 20130 islindf4 21393 scmatf1 22033 gsummatr01 22161 tdeglem4 25577 tdeglem4OLD 25578 finsumvtxdg2sstep 28806 dfconngr1 29441 satfv1lem 34353 bj-raldifsn 35981 lindsadd 36481 lindsenlbs 36483 poimirlem25 36513 poimirlem27 36515 hdmap14lem4a 40742 hdmap14lem13 40751 supxrmnf2 44143 infxrpnf2 44173 fsumnncl 44288 hoidmv1lelem2 45308 gsumdifsndf 46591 mgpsumunsn 47037 |
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