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| Description: Closed theorem form of tpid3 4773. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 30-Apr-2021.) | 
| Ref | Expression | 
|---|---|
| tpid3g | ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ 𝐴 = 𝐴 | |
| 2 | 1 | 3mix3i 1336 | . 2 ⊢ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐴) | 
| 3 | eltpg 4686 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶, 𝐷, 𝐴} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐴))) | |
| 4 | 2, 3 | mpbiri 258 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∨ w3o 1086 = wceq 1540 ∈ wcel 2108 {ctp 4630 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 df-tp 4631 | 
| This theorem is referenced by: tpid3 4773 f1dom3fv3dif 7288 f1dom3el3dif 7289 en3lplem1 9652 en3lp 9654 tpf 14538 nb3grprlem1 29397 cplgr3v 29452 cshw1s2 32945 cyc3co2 33160 en3lplem1VD 44863 en3lpVD 44865 limsupequzlem 45737 etransclem48 46297 | 
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