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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 4381 | . . 3 ⊢ {𝐶} = {𝐶, 𝐶} | |
2 | 1 | ineq2i 4009 | . 2 ⊢ ({𝐴, 𝐵} ∩ {𝐶}) = ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) |
3 | disjpr2 4438 | . . 3 ⊢ (((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅) | |
4 | 3 | anidms 563 | . 2 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅) |
5 | 2, 4 | syl5eq 2845 | 1 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ≠ wne 2971 ∩ cin 3768 ∅c0 4115 {csn 4368 {cpr 4370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-sn 4369 df-pr 4371 |
This theorem is referenced by: disjtpsn 4440 disjtp2 4441 diftpsn3 4521 funtpg 6155 funcnvtp 6163 prodtp 30091 |
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