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Mirrors > Home > MPE Home > Th. List > difprsn2 | Structured version Visualization version GIF version |
Description: Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.) |
Ref | Expression |
---|---|
difprsn2 | ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prcom 4736 | . . 3 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
2 | 1 | difeq1i 4118 | . 2 ⊢ ({𝐴, 𝐵} ∖ {𝐵}) = ({𝐵, 𝐴} ∖ {𝐵}) |
3 | necom 2994 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | |
4 | difprsn1 4803 | . . 3 ⊢ (𝐵 ≠ 𝐴 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴}) | |
5 | 3, 4 | sylbi 216 | . 2 ⊢ (𝐴 ≠ 𝐵 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴}) |
6 | 2, 5 | eqtrid 2784 | 1 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ≠ wne 2940 ∖ cdif 3945 {csn 4628 {cpr 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-nul 4323 df-sn 4629 df-pr 4631 |
This theorem is referenced by: f12dfv 7270 pmtrprfval 19354 nbgr2vtx1edg 28604 nbuhgr2vtx1edgb 28606 nfrgr2v 29522 cycpm2tr 32273 drngmxidl 32588 ldepsnlinc 47179 |
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