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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 3494 | . 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 670 = wceq 1299 ∈ wcel 1448 {cpr 3475 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 df-un 3025 df-sn 3480 df-pr 3481 |
This theorem is referenced by: nelpri 3498 nelprd 3500 opth1 4096 0nelop 4108 ontr2exmid 4378 onintexmid 4425 reg3exmidlemwe 4431 funtpg 5110 ftpg 5536 acexmidlemcase 5701 2oconcl 6266 en2eqpr 6730 eldju1st 6871 finomni 6924 exmidomniim 6925 exmidfodomrlemr 6967 exmidfodomrlemrALT 6968 sup3exmid 8573 m1expcl2 10156 maxleim 10817 maxleast 10825 zmaxcl 10835 minmax 10840 xrmaxleim 10852 xrmaxaddlem 10868 xrminmax 10873 qtopbas 12444 limcimolemlt 12513 recnprss 12529 el2oss1o 12773 nninfalllem1 12787 nninfall 12788 nninfsellemqall 12795 nninfomnilem 12798 isomninnlem 12809 trilpolemclim 12813 trilpolemcl 12814 trilpolemisumle 12815 trilpolemeq1 12817 trilpolemlt1 12818 |
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