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Mathbox for Glauco Siliprandi |
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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 961 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | |
3 | nelpr2.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | elprg 4456 | . . . . . 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 473 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) |
7 | 2, 6 | mpbird 249 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 ∈ {𝐵, 𝐶}) |
8 | 1, 7 | mtand 803 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐶) |
9 | 8 | neqned 2968 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∨ wo 833 = wceq 1507 ∈ wcel 2048 ≠ wne 2961 {cpr 4437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-v 3411 df-un 3830 df-sn 4436 df-pr 4438 |
This theorem is referenced by: ovnsubadd2lem 42304 |
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