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Mirrors > Home > MPE Home > Th. List > elpr2 | Structured version Visualization version GIF version |
Description: A member of a pair of sets is one or the other of them, and conversely. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.) (Proof shortened by JJ, 23-Jul-2021.) |
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 | elpr2.1 | . 2 ⊢ 𝐵 ∈ V | |
2 | elpr2.2 | . 2 ⊢ 𝐶 ∈ V | |
3 | elpr2g 4615 | . 2 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 846 = wceq 1542 ∈ wcel 2107 Vcvv 3448 {cpr 4593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-un 3920 df-sn 4592 df-pr 4594 |
This theorem is referenced by: elopg 5428 elxr 13044 nofv 27021 fprodex01 31763 |
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