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Mirrors > Home > MPE Home > Th. List > disjprsn | Structured version Visualization version GIF version |
Description: The disjoint intersection of an unordered pair and a singleton. (Contributed by AV, 23-Jan-2021.) |
Ref | Expression |
---|---|
disjprsn | ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 4636 | . . 3 ⊢ {𝐶} = {𝐶, 𝐶} | |
2 | 1 | ineq2i 4204 | . 2 ⊢ ({𝐴, 𝐵} ∩ {𝐶}) = ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) |
3 | disjpr2 4712 | . . 3 ⊢ (((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅) | |
4 | 3 | anidms 566 | . 2 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅) |
5 | 2, 4 | eqtrid 2778 | 1 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ≠ wne 2934 ∩ cin 3942 ∅c0 4317 {csn 4623 {cpr 4625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-sn 4624 df-pr 4626 |
This theorem is referenced by: disjtpsn 4714 disjtp2 4715 diftpsn3 4800 funtpg 6596 funcnvtp 6604 prodtp 32535 |
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