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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 4671 | . . 3 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
2 | 1 | difeq1i 4098 | . 2 ⊢ ({𝐴, 𝐵} ∖ {𝐵}) = ({𝐵, 𝐴} ∖ {𝐵}) |
3 | necom 3072 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | |
4 | difprsn1 4736 | . . 3 ⊢ (𝐵 ≠ 𝐴 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴}) | |
5 | 3, 4 | sylbi 219 | . 2 ⊢ (𝐴 ≠ 𝐵 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴}) |
6 | 2, 5 | syl5eq 2871 | 1 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ≠ wne 3019 ∖ cdif 3936 {csn 4570 {cpr 4572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-sn 4571 df-pr 4573 |
This theorem is referenced by: f12dfv 7033 pmtrprfval 18618 nbgr2vtx1edg 27135 nbuhgr2vtx1edgb 27137 nfrgr2v 28054 cycpm2tr 30765 ldepsnlinc 44570 |
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