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Mirrors > Home > MPE Home > Th. List > difprsn1 | Structured version Visualization version 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 3069 | . 2 ⊢ (𝐵 ≠ 𝐴 ↔ 𝐴 ≠ 𝐵) | |
2 | disjsn2 4648 | . . . 4 ⊢ (𝐵 ≠ 𝐴 → ({𝐵} ∩ {𝐴}) = ∅) | |
3 | disj3 4403 | . . . 4 ⊢ (({𝐵} ∩ {𝐴}) = ∅ ↔ {𝐵} = ({𝐵} ∖ {𝐴})) | |
4 | 2, 3 | sylib 220 | . . 3 ⊢ (𝐵 ≠ 𝐴 → {𝐵} = ({𝐵} ∖ {𝐴})) |
5 | df-pr 4570 | . . . . . 6 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
6 | 5 | equncomi 4131 | . . . . 5 ⊢ {𝐴, 𝐵} = ({𝐵} ∪ {𝐴}) |
7 | 6 | difeq1i 4095 | . . . 4 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) = (({𝐵} ∪ {𝐴}) ∖ {𝐴}) |
8 | difun2 4429 | . . . 4 ⊢ (({𝐵} ∪ {𝐴}) ∖ {𝐴}) = ({𝐵} ∖ {𝐴}) | |
9 | 7, 8 | eqtri 2844 | . . 3 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) = ({𝐵} ∖ {𝐴}) |
10 | 4, 9 | syl6reqr 2875 | . 2 ⊢ (𝐵 ≠ 𝐴 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵}) |
11 | 1, 10 | sylbir 237 | 1 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ≠ wne 3016 ∖ cdif 3933 ∪ cun 3934 ∩ cin 3935 ∅c0 4291 {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-ne 3017 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-sn 4568 df-pr 4570 |
This theorem is referenced by: difprsn2 4734 f12dfv 7030 pmtrprfval 18615 nbgr2vtx1edg 27132 nbuhgr2vtx1edgb 27134 nfrgr2v 28051 mptprop 30434 cycpm2tr 30761 eulerpartlemgf 31637 coinflippvt 31742 ldepsnlinc 44583 |
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