Theorem tpssi 4796

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 4587	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
2		prssi 4779	. . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
3	2	3adant3 1145	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
4		snssi 4744	. . . 4 ⊢ (𝐶 ∈ 𝐷 → {𝐶} ⊆ 𝐷)
5	4	3ad2ant3 1148	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐶} ⊆ 𝐷)
6	3, 5	unssd 4144	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
7	1, 6	eqsstrid 3974	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Syntax hints:   → wi 4   ∧ w3a 1098   ∈ wcel 2142   ∪ cun 3902   ⊆ wss 3904  {csn 4582  {cpr 4584  {ctp 4586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-un 3909  df-ss 3921  df-sn 4583  df-pr 4585  df-tp 4587

This theorem is referenced by:  sgnrn  15111  sgnclre  15115  lcmftp  16670  trgcgrg  28684  tpssd  32737  cyc3co2  33320  signstf  34860  limsupequzlem  46296  fourierdlem46  46726  fourierdlem102  46782  fourierdlem114  46794  etransclem48  46856  grtrissvtx  48566  grtrimap  48570  usgrexmpl2nb0  48653  usgrexmpl2nb3  48656