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Mirrors > Home > ILE Home > Th. List > prcom | GIF version |
Description: Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
prcom | ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 3262 | . 2 ⊢ ({𝐴} ∪ {𝐵}) = ({𝐵} ∪ {𝐴}) | |
2 | df-pr 3578 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
3 | df-pr 3578 | . 2 ⊢ {𝐵, 𝐴} = ({𝐵} ∪ {𝐴}) | |
4 | 1, 2, 3 | 3eqtr4i 2195 | 1 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ∪ cun 3110 {csn 3571 {cpr 3572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2724 df-un 3116 df-pr 3578 |
This theorem is referenced by: preq2 3649 tpcoma 3665 tpidm23 3672 prid2g 3676 prid2 3678 prprc2 3680 difprsn2 3708 preqr2g 3742 preqr2 3744 preq12b 3745 fvpr2 5685 fvpr2g 5687 en2other2 7144 maxcom 11135 mincom 11160 xrmax2sup 11185 xrmaxltsup 11189 xrmaxadd 11192 xrbdtri 11207 qtopbasss 13088 |
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