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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 4329 | . . 3 ⊢ {𝐶} = {𝐶, 𝐶} | |
2 | 1 | ineq2i 3962 | . 2 ⊢ ({𝐴, 𝐵} ∩ {𝐶}) = ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) |
3 | disjpr2 4385 | . . 3 ⊢ (((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅) | |
4 | 3 | anidms 556 | . 2 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅) |
5 | 2, 4 | syl5eq 2817 | 1 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ≠ wne 2943 ∩ cin 3722 ∅c0 4063 {csn 4316 {cpr 4318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-sn 4317 df-pr 4319 |
This theorem is referenced by: disjtpsn 4387 disjtp2 4388 diftpsn3 4468 funtpg 6082 funcnvtp 6090 prodtp 29906 |
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