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Mirrors > Home > MPE Home > Th. List > nelpr2 | Structured version Visualization version GIF version |
Description: If a class is not an element of an unordered pair, it is not the second listed element. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
nelpr2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
nelpr2.n | ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
Ref | Expression |
---|---|
nelpr2 | ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nelpr2.n | . . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶}) | |
2 | animorr 975 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
3 | nelpr2.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | elprg 4581 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) |
6 | 5 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) |
7 | 2, 6 | mpbird 259 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 ∈ {𝐵, 𝐶}) |
8 | 1, 7 | mtand 814 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐶) |
9 | 8 | neqned 3023 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 {cpr 4562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-v 3496 df-un 3940 df-sn 4561 df-pr 4563 |
This theorem is referenced by: ovnsubadd2lem 42921 |
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