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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 506 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
3 | nelpr2.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | elprg 4196 | . . . . . 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 = 𝐶) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) |
7 | 2, 6 | mpbird 247 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 ∈ {𝐵, 𝐶}) |
8 | 1, 7 | mtand 691 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐶) |
9 | 8 | neqned 2801 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 {cpr 4179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-v 3202 df-un 3579 df-sn 4178 df-pr 4180 |
This theorem is referenced by: ovnsubadd2lem 40859 |
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