Theorem tpss 3784

Description: A triplet 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 3333	. 2 ⊢ (({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷) ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
2		df-3an 982	. . 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 3774	. . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) ↔ {𝐴, 𝐵} ⊆ 𝐷)
6		tpss.3	. . . . 5 ⊢ 𝐶 ∈ V
7	6	snss 3753	. . . 4 ⊢ (𝐶 ∈ 𝐷 ↔ {𝐶} ⊆ 𝐷)
8	5, 7	anbi12i 460	. . 3 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) ∧ 𝐶 ∈ 𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷))
9	2, 8	bitri 184	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷))
10		df-tp 3626	. . 3 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
11	10	sseq1i 3205	. 2 ⊢ ({𝐴, 𝐵, 𝐶} ⊆ 𝐷 ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
12	1, 9, 11	3bitr4i 212	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Syntax hints:   ∧ wa 104   ↔ wb 105   ∧ w3a 980   ∈ wcel 2164  Vcvv 2760   ∪ cun 3151   ⊆ wss 3153  {csn 3618  {cpr 3619  {ctp 3620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-tp 3626

This theorem is referenced by: (None)