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| 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 4631 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
| 2 | prssi 4821 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ⊆ 𝐷) | |
| 3 | 2 | 3adant3 1133 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵} ⊆ 𝐷) | 
| 4 | snssi 4808 | . . . 4 ⊢ (𝐶 ∈ 𝐷 → {𝐶} ⊆ 𝐷) | |
| 5 | 4 | 3ad2ant3 1136 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐶} ⊆ 𝐷) | 
| 6 | 3, 5 | unssd 4192 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷) | 
| 7 | 1, 6 | eqsstrid 4022 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1087 ∈ wcel 2108 ∪ cun 3949 ⊆ wss 3951 {csn 4626 {cpr 4628 {ctp 4630 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 df-tp 4631 | 
| This theorem is referenced by: lcmftp 16673 trgcgrg 28523 cyc3co2 33160 sgnclre 34542 signstf 34581 limsupequzlem 45737 fourierdlem46 46167 fourierdlem102 46223 fourierdlem114 46235 etransclem48 46297 grtrissvtx 47911 grtrimap 47915 usgrexmpl2nb0 47990 usgrexmpl2nb3 47993 | 
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