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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 4582 | . . 3 ⊢ {𝐶} = {𝐶, 𝐶} | |
2 | 1 | ineq2i 4188 | . 2 ⊢ ({𝐴, 𝐵} ∩ {𝐶}) = ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) |
3 | disjpr2 4651 | . . 3 ⊢ (((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅) | |
4 | 3 | anidms 569 | . 2 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅) |
5 | 2, 4 | syl5eq 2870 | 1 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ≠ wne 3018 ∩ cin 3937 ∅c0 4293 {csn 4569 {cpr 4571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-sn 4570 df-pr 4572 |
This theorem is referenced by: disjtpsn 4653 disjtp2 4654 diftpsn3 4737 funtpg 6411 funcnvtp 6419 prodtp 30545 |
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