Theorem tpssi 4831

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 4625	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
2		prssi 4816	. . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
3	2	3adant3 1129	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
4		snssi 4803	. . . 4 ⊢ (𝐶 ∈ 𝐷 → {𝐶} ⊆ 𝐷)
5	4	3ad2ant3 1132	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐶} ⊆ 𝐷)
6	3, 5	unssd 4178	. 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)

Syntax hints:   → wi 4   ∧ w3a 1084   ∈ wcel 2098   ∪ cun 3938   ⊆ wss 3940  {csn 4620  {cpr 4622  {ctp 4624

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 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-un 3945  df-in 3947  df-ss 3957  df-sn 4621  df-pr 4623  df-tp 4625

This theorem is referenced by:  lcmftp  16570  trgcgrg  28235  cyc3co2  32767  sgnclre  34027  signstf  34066  limsupequzlem  44923  fourierdlem46  45353  fourierdlem102  45409  fourierdlem114  45421  etransclem48  45483