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Mirrors > Home > ILE Home > Th. List > elpri | GIF version |
Description: If a class is an element of a pair, then it is one of the two paired elements. (Contributed by Scott Fenton, 1-Apr-2011.) |
Ref | Expression |
---|---|
elpri | ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elprg 3517 | . 2 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) | |
2 | 1 | ibi 175 | 1 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 682 = wceq 1316 ∈ wcel 1465 {cpr 3498 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 |
This theorem is referenced by: nelpri 3521 nelprd 3523 opth1 4128 0nelop 4140 ontr2exmid 4410 onintexmid 4457 reg3exmidlemwe 4463 funtpg 5144 ftpg 5572 acexmidlemcase 5737 2oconcl 6304 en2eqpr 6769 eldju1st 6924 finomni 6980 exmidomniim 6981 ismkvnex 6997 exmidonfinlem 7017 exmidfodomrlemr 7026 exmidfodomrlemrALT 7027 exmidaclem 7032 sup3exmid 8683 m1expcl2 10283 maxleim 10945 maxleast 10953 zmaxcl 10964 minmax 10969 xrmaxleim 10981 xrmaxaddlem 10997 xrminmax 11002 unct 11881 qtopbas 12618 limcimolemlt 12729 recnprss 12752 coseq0negpitopi 12844 el2oss1o 13115 nninfalllem1 13130 nninfall 13131 nninfsellemqall 13138 nninfomnilem 13141 isomninnlem 13152 trilpolemclim 13156 trilpolemcl 13157 trilpolemisumle 13158 trilpolemeq1 13160 trilpolemlt1 13161 |
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