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Mirrors > Home > NFE Home > Th. List > elpri | GIF version |
Description: If a class is an element of a pair, then it is one of the two paired elements. (Contributed by Scott Fenton, 1-Apr-2011.) |
Ref | Expression |
---|---|
elpri | ⊢ (A ∈ {B, C} → (A = B ∨ A = C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elprg 3750 | . 2 ⊢ (A ∈ {B, C} → (A ∈ {B, C} ↔ (A = B ∨ A = C))) | |
2 | 1 | ibi 232 | 1 ⊢ (A ∈ {B, C} → (A = B ∨ A = C)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 357 = wceq 1642 ∈ wcel 1710 {cpr 3738 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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: nelpri 3754 |
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