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Mirrors > Home > MPE Home > Th. List > nelpr1 | Structured version Visualization version GIF version |
Description: If a class is not an element of an unordered pair, it is not the first listed element. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
nelpr1.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
nelpr1.n | ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶}) |
Ref | Expression |
---|---|
nelpr1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nelpr1.n | . . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶}) | |
2 | animorrl 973 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
3 | nelpr1.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | elprg 4487 | . . . . . 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 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) |
7 | 2, 6 | mpbird 258 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ {𝐵, 𝐶}) |
8 | 1, 7 | mtand 812 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
9 | 8 | neqned 2989 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 842 = wceq 1520 ∈ wcel 2079 ≠ wne 2982 {cpr 4468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-ext 2767 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-v 3434 df-un 3859 df-sn 4467 df-pr 4469 |
This theorem is referenced by: cyc3genpmlem 30389 ovnsubadd2lem 42423 |
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