Theorem tpssi 4787

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 4578	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
2		prssi 4770	. . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
3	2	3adant3 1132	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵} ⊆ 𝐷)
4		snssi 4757	. . . 4 ⊢ (𝐶 ∈ 𝐷 → {𝐶} ⊆ 𝐷)
5	4	3ad2ant3 1135	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐶} ⊆ 𝐷)
6	3, 5	unssd 4139	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
7	1, 6	eqsstrid 3968	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Syntax hints:   → wi 4   ∧ w3a 1086   ∈ wcel 2111   ∪ cun 3895   ⊆ wss 3897  {csn 4573  {cpr 4575  {ctp 4577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3902  df-ss 3914  df-sn 4574  df-pr 4576  df-tp 4578

This theorem is referenced by:  lcmftp  16547  trgcgrg  28493  tpssd  32518  sgnclre  32815  cyc3co2  33109  signstf  34579  limsupequzlem  45830  fourierdlem46  46260  fourierdlem102  46316  fourierdlem114  46328  etransclem48  46390  grtrissvtx  48054  grtrimap  48058  usgrexmpl2nb0  48141  usgrexmpl2nb3  48144