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Mirrors > Home > MPE Home > Th. List > eltpg | Structured version Visualization version GIF version |
Description: Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
eltpg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elprg 4649 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) | |
2 | elsng 4642 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐷} ↔ 𝐴 = 𝐷)) | |
3 | 1, 2 | orbi12d 916 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷}) ↔ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ∨ 𝐴 = 𝐷))) |
4 | df-tp 4633 | . . . 4 ⊢ {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷}) | |
5 | 4 | eleq2i 2824 | . . 3 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ 𝐴 ∈ ({𝐵, 𝐶} ∪ {𝐷})) |
6 | elun 4148 | . . 3 ⊢ (𝐴 ∈ ({𝐵, 𝐶} ∪ {𝐷}) ↔ (𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷})) | |
7 | 5, 6 | bitri 275 | . 2 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 ∈ {𝐵, 𝐶} ∨ 𝐴 ∈ {𝐷})) |
8 | df-3or 1087 | . 2 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ↔ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ∨ 𝐴 = 𝐷)) | |
9 | 3, 7, 8 | 3bitr4g 314 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∨ wo 844 ∨ w3o 1085 = wceq 1540 ∈ wcel 2105 ∪ cun 3946 {csn 4628 {cpr 4630 {ctp 4632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-un 3953 df-sn 4629 df-pr 4631 df-tp 4633 |
This theorem is referenced by: eldiftp 4690 eltpi 4691 eltp 4692 tpid1g 4773 tpid2g 4775 tpid3g 4776 f1dom3fv3dif 7270 f1dom3el3dif 7271 lcmftp 16580 estrreslem2 18100 1cubr 26688 zabsle1 27143 nb3grprlem1 29071 |
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