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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 4138 | . . 3 ⊢ {𝐶} = {𝐶, 𝐶} | |
2 | 1 | ineq2i 3773 | . 2 ⊢ ({𝐴, 𝐵} ∩ {𝐶}) = ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) |
3 | disjpr2 4194 | . . 3 ⊢ (((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅) | |
4 | 3 | anidms 675 | . 2 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅) |
5 | 2, 4 | syl5eq 2656 | 1 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ≠ wne 2780 ∩ cin 3539 ∅c0 3874 {csn 4125 {cpr 4127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 df-pr 4128 |
This theorem is referenced by: diftpsn3 4273 funtpg 5856 funcnvtp 5865 |
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