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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 4651 | . 2 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∨ wo 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {cpr 4628 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 | 
| This theorem is referenced by: elopg 5471 elxr 13158 nofv 27702 fprodex01 32827 opgpgvtx 48010 | 
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