Theorem tpss 4837

Description: An unordered triple of elements of a class is a subset of the class. (Contributed by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Hypotheses

Ref	Expression
tpss.1	⊢ 𝐴 ∈ V
tpss.2	⊢ 𝐵 ∈ V
tpss.3	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
tpss	⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Proof of Theorem tpss

Step	Hyp	Ref	Expression
1		unss 4183	. 2 ⊢ (({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷) ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
2		df-3an 1089	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) ∧ 𝐶 ∈ 𝐷))
3		tpss.1	. . . . 5 ⊢ 𝐴 ∈ V
4		tpss.2	. . . . 5 ⊢ 𝐵 ∈ V
5	3, 4	prss 4822	. . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) ↔ {𝐴, 𝐵} ⊆ 𝐷)
6		tpss.3	. . . . 5 ⊢ 𝐶 ∈ V
7	6	snss 4788	. . . 4 ⊢ (𝐶 ∈ 𝐷 ↔ {𝐶} ⊆ 𝐷)
8	5, 7	anbi12i 627	. . 3 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) ∧ 𝐶 ∈ 𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷))
9	2, 8	bitri 274	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷))
10		df-tp 4632	. . 3 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
11	10	sseq1i 4009	. 2 ⊢ ({𝐴, 𝐵, 𝐶} ⊆ 𝐷 ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
12	1, 9, 11	3bitr4i 302	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   ∈ wcel 2106  Vcvv 3474   ∪ cun 3945   ⊆ wss 3947  {csn 4627  {cpr 4629  {ctp 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-un 3952  df-in 3954  df-ss 3964  df-sn 4628  df-pr 4630  df-tp 4632

This theorem is referenced by:  1cubr  26336  konigsberglem4  29497  rabren3dioph  41538  fourierdlem102  44910  fourierdlem114  44922  nnsum4primesodd  46450  nnsum4primesoddALTV  46451