Theorem prssg 3863

Description: A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 22-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

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

Proof of Theorem prssg

Step	Hyp	Ref	Expression
1		snssg 3845	. . 3 ⊢ (A ∈ V → (A ∈ C ↔ {A} ⊆ C))
2		snssg 3845	. . 3 ⊢ (B ∈ W → (B ∈ C ↔ {B} ⊆ C))
3	1, 2	bi2anan9 843	. 2 ⊢ ((A ∈ V ∧ B ∈ W) → ((A ∈ C ∧ B ∈ C) ↔ ({A} ⊆ C ∧ {B} ⊆ C)))
4		unss 3438	. . 3 ⊢ (({A} ⊆ C ∧ {B} ⊆ C) ↔ ({A} ∪ {B}) ⊆ C)
5		df-pr 3743	. . . 4 ⊢ {A, B} = ({A} ∪ {B})
6	5	sseq1i 3296	. . 3 ⊢ ({A, B} ⊆ C ↔ ({A} ∪ {B}) ⊆ C)
7	4, 6	bitr4i 243	. 2 ⊢ (({A} ⊆ C ∧ {B} ⊆ C) ↔ {A, B} ⊆ C)
8	3, 7	syl6bb 252	1 ⊢ ((A ∈ V ∧ B ∈ W) → ((A ∈ C ∧ B ∈ C) ↔ {A, B} ⊆ C))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∈ wcel 1710   ∪ cun 3208   ⊆ wss 3258  {csn 3738  {cpr 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-ss 3260  df-sn 3742  df-pr 3743

This theorem is referenced by:  prssi  3864