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Mirrors > Home > MPE Home > Th. List > disjtpsn | Structured version Visualization version GIF version |
Description: The disjoint intersection of an unordered triple and a singleton. (Contributed by AV, 14-Nov-2021.) |
Ref | Expression |
---|---|
disjtpsn | ⊢ ((𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷 ∧ 𝐶 ≠ 𝐷) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷}) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 4632 | . . 3 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
2 | 1 | ineq1i 4207 | . 2 ⊢ ({𝐴, 𝐵, 𝐶} ∩ {𝐷}) = (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷}) |
3 | disjprsn 4717 | . . . . 5 ⊢ ((𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷) → ({𝐴, 𝐵} ∩ {𝐷}) = ∅) | |
4 | 3 | 3adant3 1130 | . . . 4 ⊢ ((𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷 ∧ 𝐶 ≠ 𝐷) → ({𝐴, 𝐵} ∩ {𝐷}) = ∅) |
5 | disjsn2 4715 | . . . . 5 ⊢ (𝐶 ≠ 𝐷 → ({𝐶} ∩ {𝐷}) = ∅) | |
6 | 5 | 3ad2ant3 1133 | . . . 4 ⊢ ((𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷 ∧ 𝐶 ≠ 𝐷) → ({𝐶} ∩ {𝐷}) = ∅) |
7 | 4, 6 | jca 510 | . . 3 ⊢ ((𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷 ∧ 𝐶 ≠ 𝐷) → (({𝐴, 𝐵} ∩ {𝐷}) = ∅ ∧ ({𝐶} ∩ {𝐷}) = ∅)) |
8 | undisj1 4460 | . . 3 ⊢ ((({𝐴, 𝐵} ∩ {𝐷}) = ∅ ∧ ({𝐶} ∩ {𝐷}) = ∅) ↔ (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷}) = ∅) | |
9 | 7, 8 | sylib 217 | . 2 ⊢ ((𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷 ∧ 𝐶 ≠ 𝐷) → (({𝐴, 𝐵} ∪ {𝐶}) ∩ {𝐷}) = ∅) |
10 | 2, 9 | eqtrid 2782 | 1 ⊢ ((𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷 ∧ 𝐶 ≠ 𝐷) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷}) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ≠ wne 2938 ∪ cun 3945 ∩ cin 3946 ∅c0 4321 {csn 4627 {cpr 4629 {ctp 4631 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-sn 4628 df-pr 4630 df-tp 4632 |
This theorem is referenced by: disjtp2 4719 cnfldfunALT 21157 cnfldfunALTOLD 21158 gg-cnfldfunALT 35484 |
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