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Mirrors > Home > MPE Home > Th. List > elpr2OLD | Structured version Visualization version GIF version |
Description: Obsolete version of elpr2 4540 as of 25-May-2024. (Contributed by NM, 14-Oct-2005.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elpr2.1 | ⊢ 𝐵 ∈ V |
elpr2.2 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
elpr2OLD | ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3427 | . 2 ⊢ (𝐴 ∈ {𝐵, 𝐶} → 𝐴 ∈ V) | |
2 | elpr2.1 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | eleq1 2838 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V)) | |
4 | 2, 3 | mpbiri 261 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ∈ V) |
5 | elpr2.2 | . . . 4 ⊢ 𝐶 ∈ V | |
6 | eleq1 2838 | . . . 4 ⊢ (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V)) | |
7 | 5, 6 | mpbiri 261 | . . 3 ⊢ (𝐴 = 𝐶 → 𝐴 ∈ V) |
8 | 4, 7 | jaoi 855 | . 2 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → 𝐴 ∈ V) |
9 | elprg 4536 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) | |
10 | 1, 8, 9 | pm5.21nii 384 | 1 ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∨ wo 845 = wceq 1539 ∈ wcel 2112 Vcvv 3407 {cpr 4517 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-tru 1542 df-ex 1783 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-v 3409 df-un 3859 df-sn 4516 df-pr 4518 |
This theorem is referenced by: (None) |
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