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Mirrors > Home > MPE Home > Th. List > tpssi | Structured version Visualization version GIF version |
Description: An unordered triple of elements of a class is a subset of the class. (Contributed by Alexander van der Vekens, 1-Feb-2018.) |
Ref | Expression |
---|---|
tpssi | ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 4636 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
2 | prssi 4826 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ⊆ 𝐷) | |
3 | 2 | 3adant3 1131 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵} ⊆ 𝐷) |
4 | snssi 4813 | . . . 4 ⊢ (𝐶 ∈ 𝐷 → {𝐶} ⊆ 𝐷) | |
5 | 4 | 3ad2ant3 1134 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐶} ⊆ 𝐷) |
6 | 3, 5 | unssd 4202 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷) |
7 | 1, 6 | eqsstrid 4044 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 ∈ wcel 2106 ∪ cun 3961 ⊆ wss 3963 {csn 4631 {cpr 4633 {ctp 4635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-ss 3980 df-sn 4632 df-pr 4634 df-tp 4636 |
This theorem is referenced by: lcmftp 16670 trgcgrg 28538 cyc3co2 33143 sgnclre 34521 signstf 34560 limsupequzlem 45678 fourierdlem46 46108 fourierdlem102 46164 fourierdlem114 46176 etransclem48 46238 grtrissvtx 47849 grtrimap 47851 usgrexmpl2nb0 47926 usgrexmpl2nb3 47929 |
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