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Mirrors > Home > ILE Home > Th. List > elpr | GIF version |
Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.) |
Ref | Expression |
---|---|
elpr.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elpr | ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpr.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | elprg 3581 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∨ wo 698 = wceq 1335 ∈ wcel 2128 Vcvv 2712 {cpr 3562 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-un 3106 df-sn 3567 df-pr 3568 |
This theorem is referenced by: prmg 3682 difprsnss 3696 preqr1 3733 preq12b 3735 prel12 3736 pwprss 3770 pwtpss 3771 unipr 3788 intpr 3841 zfpair2 4172 elop 4193 ordtri2or2exmidlem 4487 onsucelsucexmidlem 4490 en2lp 4515 reg3exmidlemwe 4540 xpsspw 4700 acexmidlem2 5823 2oconcl 6388 exmidpw 6855 exmidpweq 6856 renfdisj 7939 fzpr 9985 maxabslemval 11119 xrmaxiflemval 11158 isprm2 12009 bj-zfpair2 13556 ss1oel2o 13636 |
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