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 e. V /\ B e. W) -> ((A e. C /\ B e. C) <-> {A, B} C_ C))

Proof of Theorem prssg

Step	Hyp	Ref	Expression
1		snssg 3845	. . 3 |- (A e. V -> (A e. C <-> {A} C_ C))
2		snssg 3845	. . 3 |- (B e. W -> (B e. C <-> {B} C_ C))
3	1, 2	bi2anan9 843	. 2 |- ((A e. V /\ B e. W) -> ((A e. C /\ B e. C) <-> ({A} C_ C /\ {B} C_ C)))
4		unss 3438	. . 3 |- (({A} C_ C /\ {B} C_ C) <-> ({A} u. {B}) C_ C)
5		df-pr 3743	. . . 4 |- {A, B} = ({A} u. {B})
6	5	sseq1i 3296	. . 3 |- ({A, B} C_ C <-> ({A} u. {B}) C_ C)
7	4, 6	bitr4i 243	. 2 |- (({A} C_ C /\ {B} C_ C) <-> {A, B} C_ C)
8	3, 7	syl6bb 252	1 |- ((A e. V /\ B e. W) -> ((A e. C /\ B e. C) <-> {A, B} C_ C))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   e. wcel 1710   u. cun 3208   C_ 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