Theorem prss 3861

Description: A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Hypotheses

Ref	Expression
prss.1	⊢ A ∈ V
prss.2	⊢ B ∈ V

Assertion

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

Proof of Theorem prss

Step	Hyp	Ref	Expression
1		unss 3437	. 2 ⊢ (({A} ⊆ C ∧ {B} ⊆ C) ↔ ({A} ∪ {B}) ⊆ C)
2		prss.1	. . . 4 ⊢ A ∈ V
3	2	snss 3838	. . 3 ⊢ (A ∈ C ↔ {A} ⊆ C)
4		prss.2	. . . 4 ⊢ B ∈ V
5	4	snss 3838	. . 3 ⊢ (B ∈ C ↔ {B} ⊆ C)
6	3, 5	anbi12i 678	. 2 ⊢ ((A ∈ C ∧ B ∈ C) ↔ ({A} ⊆ C ∧ {B} ⊆ C))
7		df-pr 3742	. . 3 ⊢ {A, B} = ({A} ∪ {B})
8	7	sseq1i 3295	. 2 ⊢ ({A, B} ⊆ C ↔ ({A} ∪ {B}) ⊆ C)
9	1, 6, 8	3bitr4i 268	1 ⊢ ((A ∈ C ∧ B ∈ C) ↔ {A, B} ⊆ C)

Syntax hints:   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  Vcvv 2859   ∪ cun 3207   ⊆ wss 3257  {csn 3737  {cpr 3738

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-ss 3259  df-sn 3741  df-pr 3742

This theorem is referenced by:  tpss  3871  prsspw  3878  uniintsn  3963