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Mirrors > Home > MPE Home > Th. List > difpr | Structured version Visualization version GIF version |
Description: Removing two elements as pair of elements corresponds to removing each of the two elements as singletons. (Contributed by Alexander van der Vekens, 13-Jul-2018.) |
Ref | Expression |
---|---|
difpr | ⊢ (𝐴 ∖ {𝐵, 𝐶}) = ((𝐴 ∖ {𝐵}) ∖ {𝐶}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4570 | . . 3 ⊢ {𝐵, 𝐶} = ({𝐵} ∪ {𝐶}) | |
2 | 1 | difeq2i 4096 | . 2 ⊢ (𝐴 ∖ {𝐵, 𝐶}) = (𝐴 ∖ ({𝐵} ∪ {𝐶})) |
3 | difun1 4264 | . 2 ⊢ (𝐴 ∖ ({𝐵} ∪ {𝐶})) = ((𝐴 ∖ {𝐵}) ∖ {𝐶}) | |
4 | 2, 3 | eqtri 2844 | 1 ⊢ (𝐴 ∖ {𝐵, 𝐶}) = ((𝐴 ∖ {𝐵}) ∖ {𝐶}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∖ cdif 3933 ∪ cun 3934 {csn 4567 {cpr 4569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-pr 4570 |
This theorem is referenced by: hashdifpr 13777 nbgrssvwo2 27144 nbupgrres 27146 nbupgruvtxres 27189 uvtxupgrres 27190 pmtrcnelor 30735 |
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