Theorem tpssi 4863

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 4653	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
2		prssi 4846	. . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
3	2	3adant3 1132	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
4		snssi 4833	. . . 4 ⊢ (𝐶 ∈ 𝐷 → {𝐶} ⊆ 𝐷)
5	4	3ad2ant3 1135	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐶} ⊆ 𝐷)
6	3, 5	unssd 4215	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
7	1, 6	eqsstrid 4057	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Syntax hints:   → wi 4   ∧ w3a 1087   ∈ wcel 2108   ∪ cun 3974   ⊆ wss 3976  {csn 4648  {cpr 4650  {ctp 4652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-un 3981  df-ss 3993  df-sn 4649  df-pr 4651  df-tp 4653

This theorem is referenced by:  lcmftp  16683  trgcgrg  28541  cyc3co2  33133  sgnclre  34504  signstf  34543  limsupequzlem  45643  fourierdlem46  46073  fourierdlem102  46129  fourierdlem114  46141  etransclem48  46203  grtrissvtx  47795  grtrimap  47797  usgrexmpl2nb0  47846  usgrexmpl2nb3  47849