Theorem tpssg 32473

Description: An unordered triple of elements of a class is a subset of the class. (Contributed by Thierry Arnoux, 2-Nov-2025.)

Assertion

Ref	Expression
tpssg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷))

Proof of Theorem tpssg

Step	Hyp	Ref	Expression
1		df-3an 1088	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) ∧ 𝐶 ∈ 𝐷))
2		prssg 4791	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) ↔ {𝐴, 𝐵} ⊆ 𝐷))
3		snssg 4755	. . . . 5 ⊢ (𝐶 ∈ 𝑋 → (𝐶 ∈ 𝐷 ↔ {𝐶} ⊆ 𝐷))
4	2, 3	bi2anan9 638	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) ∧ 𝐶 ∈ 𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷)))
5		unss 4161	. . . . 5 ⊢ (({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷) ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
6		df-tp 4602	. . . . . 6 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
7	6	sseq1i 3983	. . . . 5 ⊢ ({𝐴, 𝐵, 𝐶} ⊆ 𝐷 ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
8	5, 7	bitr4i 278	. . . 4 ⊢ (({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
9	4, 8	bitrdi 287	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) ∧ 𝐶 ∈ 𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷))
10	1, 9	bitrid 283	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷))
11	10	3impa 1109	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2109   ∪ cun 3920   ⊆ wss 3922  {csn 4597  {cpr 4599  {ctp 4601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3457  df-un 3927  df-ss 3939  df-sn 4598  df-pr 4600  df-tp 4602

This theorem is referenced by:  tpssad  32475