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Mirrors > Home > MPE Home > Th. List > prneprprc | Structured version Visualization version GIF version |
Description: A proper unordered pair is not an improper unordered pair. (Contributed by AV, 13-Jun-2022.) |
Ref | Expression |
---|---|
prneprprc | ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prnesn 4865 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≠ {𝐷}) | |
2 | 1 | adantr 480 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐷}) |
3 | prprc1 4770 | . . . 4 ⊢ (¬ 𝐶 ∈ V → {𝐶, 𝐷} = {𝐷}) | |
4 | 3 | neeq2d 2999 | . . 3 ⊢ (¬ 𝐶 ∈ V → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} ↔ {𝐴, 𝐵} ≠ {𝐷})) |
5 | 4 | adantl 481 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ ¬ 𝐶 ∈ V) → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} ↔ {𝐴, 𝐵} ≠ {𝐷})) |
6 | 2, 5 | mpbird 257 | 1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 {csn 4631 {cpr 4633 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-v 3480 df-dif 3966 df-un 3968 df-nul 4340 df-sn 4632 df-pr 4634 |
This theorem is referenced by: preq12nebg 4868 opthprneg 4870 |
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