Theorem tpss 3651

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 3216	. 2 ⊢ (({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷) ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
2		df-3an 947	. . 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 3642	. . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) ↔ {𝐴, 𝐵} ⊆ 𝐷)
6		tpss.3	. . . . 5 ⊢ 𝐶 ∈ V
7	6	snss 3615	. . . 4 ⊢ (𝐶 ∈ 𝐷 ↔ {𝐶} ⊆ 𝐷)
8	5, 7	anbi12i 453	. . 3 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷) ∧ 𝐶 ∈ 𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷))
9	2, 8	bitri 183	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ ({𝐴, 𝐵} ⊆ 𝐷 ∧ {𝐶} ⊆ 𝐷))
10		df-tp 3501	. . 3 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
11	10	sseq1i 3089	. 2 ⊢ ({𝐴, 𝐵, 𝐶} ⊆ 𝐷 ↔ ({𝐴, 𝐵} ∪ {𝐶}) ⊆ 𝐷)
12	1, 9, 11	3bitr4i 211	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Syntax hints:   ∧ wa 103   ↔ wb 104   ∧ w3a 945   ∈ wcel 1463  Vcvv 2657   ∪ cun 3035   ⊆ wss 3037  {csn 3493  {cpr 3494  {ctp 3495

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-sn 3499  df-pr 3500  df-tp 3501

This theorem is referenced by: (None)