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Mirrors > Home > MPE Home > Th. List > eltp | Structured version Visualization version GIF version |
Description: A member of an unordered triple of classes is one of them. Special case of Exercise 1 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Apr-1994.) (Revised by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
eltp.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
eltp | ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eltp.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | eltpg 4174 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∨ w3o 1030 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {ctp 4129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-sn 4126 df-pr 4128 df-tp 4130 |
This theorem is referenced by: dftp2 4178 tpid1 4246 tpid2 4247 tpres 6371 fntpb 6378 bpoly3 14628 gausslemma2dlem0i 24889 2lgsoddprm 24941 nb3graprlem1 25980 frgra3vlem1 26527 frgra3vlem2 26528 brtp 30892 sltsolem1 31067 fmtno4prmfac 40022 nb3grprlem1 40608 frgr3vlem1 41443 frgr3vlem2 41444 |
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