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Mirrors > Home > NFE Home > Th. List > elpr2 | 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, 14-Oct-2005.) |
Ref | Expression |
---|---|
elpr2.1 | ⊢ B ∈ V |
elpr2.2 | ⊢ C ∈ V |
Ref | Expression |
---|---|
elpr2 | ⊢ (A ∈ {B, C} ↔ (A = B ∨ A = C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elprg 3750 | . . 3 ⊢ (A ∈ {B, C} → (A ∈ {B, C} ↔ (A = B ∨ A = C))) | |
2 | 1 | ibi 232 | . 2 ⊢ (A ∈ {B, C} → (A = B ∨ A = C)) |
3 | elpr2.1 | . . . . . 6 ⊢ B ∈ V | |
4 | eleq1 2413 | . . . . . 6 ⊢ (A = B → (A ∈ V ↔ B ∈ V)) | |
5 | 3, 4 | mpbiri 224 | . . . . 5 ⊢ (A = B → A ∈ V) |
6 | elpr2.2 | . . . . . 6 ⊢ C ∈ V | |
7 | eleq1 2413 | . . . . . 6 ⊢ (A = C → (A ∈ V ↔ C ∈ V)) | |
8 | 6, 7 | mpbiri 224 | . . . . 5 ⊢ (A = C → A ∈ V) |
9 | 5, 8 | jaoi 368 | . . . 4 ⊢ ((A = B ∨ A = C) → A ∈ V) |
10 | elprg 3750 | . . . 4 ⊢ (A ∈ V → (A ∈ {B, C} ↔ (A = B ∨ A = C))) | |
11 | 9, 10 | syl 15 | . . 3 ⊢ ((A = B ∨ A = C) → (A ∈ {B, C} ↔ (A = B ∨ A = C))) |
12 | 11 | ibir 233 | . 2 ⊢ ((A = B ∨ A = C) → A ∈ {B, C}) |
13 | 2, 12 | impbii 180 | 1 ⊢ (A ∈ {B, C} ↔ (A = B ∨ A = C)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∨ wo 357 = wceq 1642 ∈ wcel 1710 Vcvv 2859 {cpr 3738 |
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-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 |
This theorem is referenced by: elopk 4129 |
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