New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > eltpg | 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 3750 | . . 3 ⊢ (A ∈ V → (A ∈ {B, C} ↔ (A = B ∨ A = C))) | |
2 | elsncg 3755 | . . 3 ⊢ (A ∈ V → (A ∈ {D} ↔ A = D)) | |
3 | 1, 2 | orbi12d 690 | . 2 ⊢ (A ∈ V → ((A ∈ {B, C} ∨ A ∈ {D}) ↔ ((A = B ∨ A = C) ∨ A = D))) |
4 | df-tp 3743 | . . . 4 ⊢ {B, C, D} = ({B, C} ∪ {D}) | |
5 | 4 | eleq2i 2417 | . . 3 ⊢ (A ∈ {B, C, D} ↔ A ∈ ({B, C} ∪ {D})) |
6 | elun 3220 | . . 3 ⊢ (A ∈ ({B, C} ∪ {D}) ↔ (A ∈ {B, C} ∨ A ∈ {D})) | |
7 | 5, 6 | bitri 240 | . 2 ⊢ (A ∈ {B, C, D} ↔ (A ∈ {B, C} ∨ A ∈ {D})) |
8 | df-3or 935 | . 2 ⊢ ((A = B ∨ A = C ∨ A = D) ↔ ((A = B ∨ A = C) ∨ A = D)) | |
9 | 3, 7, 8 | 3bitr4g 279 | 1 ⊢ (A ∈ V → (A ∈ {B, C, D} ↔ (A = B ∨ A = C ∨ A = D))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∨ wo 357 ∨ w3o 933 = wceq 1642 ∈ wcel 1710 ∪ cun 3207 {csn 3737 {cpr 3738 {ctp 3739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 df-sn 3741 df-pr 3742 df-tp 3743 |
This theorem is referenced by: eltpi 3770 eltp 3771 |
Copyright terms: Public domain | W3C validator |