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Mirrors > Home > MPE Home > Th. List > prnesn | Structured version Visualization version GIF version |
Description: A proper unordered pair is not a (proper or improper) singleton. (Contributed by AV, 13-Jun-2022.) |
Ref | Expression |
---|---|
prnesn | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≠ {𝐶}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqtr3 2763 | . . . 4 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵) | |
2 | 1 | necon3ai 2965 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) |
3 | 2 | 3ad2ant3 1137 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) |
4 | simp1 1138 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑉) | |
5 | simp2 1139 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑊) | |
6 | 4, 5 | preqsnd 4769 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))) |
7 | 6 | necon3abid 2977 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ≠ {𝐶} ↔ ¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))) |
8 | 3, 7 | mpbird 260 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≠ {𝐶}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 {csn 4541 {cpr 4543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3410 df-dif 3869 df-un 3871 df-nul 4238 df-sn 4542 df-pr 4544 |
This theorem is referenced by: prneprprc 4771 |
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