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| 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 4639 | . . 3 ⊢ {𝐶} = {𝐶, 𝐶} | |
| 2 | 1 | ineq2i 4217 | . 2 ⊢ ({𝐴, 𝐵} ∩ {𝐶}) = ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) | 
| 3 | disjpr2 4713 | . . 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 2789 | 1 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ≠ wne 2940 ∩ cin 3950 ∅c0 4333 {csn 4626 {cpr 4628 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-sn 4627 df-pr 4629 | 
| This theorem is referenced by: disjtpsn 4715 disjtp2 4716 diftpsn3 4802 funtpg 6621 funcnvtp 6629 hash3tpexb 14533 prodtp 32829 gsumtp 33061 | 
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