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Theorem tpssi 4840

Description: An unordered triple of elements of a class is a subset of the class. (Contributed by Alexander van der Vekens, 1-Feb-2018.)

**Assertion**
Ref	Expression
tpssi	⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

**Proof of Theorem tpssi**
Step	Hyp	Ref	Expression
1		df-tp 4634	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
2		prssi 4825	. . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
3	2	3adant3 1129	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
4		snssi 4812	. . . 4 ⊢ (𝐶 ∈ 𝐷 → {𝐶} ⊆ 𝐷)
5	4	3ad2ant3 1132	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐶} ⊆ 𝐷)
6	3, 5	unssd 4185	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
7	1, 6	eqsstrid 4026	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1084 ∈ wcel 2098 ∪ cun 3943 ⊆ wss 3945 {csn 4629 {cpr 4631 {ctp 4633

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3465 df-un 3950 df-ss 3962 df-sn 4630 df-pr 4632 df-tp 4634

This theorem is referenced by: lcmftp 16607 trgcgrg 28382 cyc3co2 32935 sgnclre 34246 signstf 34285 limsupequzlem 45190 fourierdlem46 45620 fourierdlem102 45676 fourierdlem114 45688 etransclem48 45750

	
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