Theorem prss 1854

Description: A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49.

Hypotheses

Ref	Expression
prss.1	|- A e. V
prss.2	|- B e. V

Assertion

Ref	Expression
prss	|- ((A e. C /\ B e. C) <-> {A, B} (_ C)

Proof of Theorem prss

Step	Hyp	Ref	Expression
1		eleq1a 1158	. . . . 5 |- (A e. C -> (x = A -> x e. C))
2		eleq1a 1158	. . . . 5 |- (B e. C -> (x = B -> x e. C))
3	1, 2	jaao 330	. . . 4 |- ((A e. C /\ B e. C) -> ((x = A \/ x = B) -> x e. C))
4		visset 1350	. . . . 5 |- x e. V
5	4	elpr 1823	. . . 4 |- (x e. {A, B} <-> (x = A \/ x = B))
6	3, 5	syl5ib 181	. . 3 |- ((A e. C /\ B e. C) -> (x e. {A, B} -> x e. C))
7	6	ssrdv 1509	. 2 |- ((A e. C /\ B e. C) -> {A, B} (_ C)
8		prss.1	. . . . 5 |- A e. V
9	8	pri1 1841	. . . 4 |- A e. {A, B}
10		ssel 1502	. . . 4 |- ({A, B} (_ C -> (A e. {A, B} -> A e. C))
11	9, 10	mpi 44	. . 3 |- ({A, B} (_ C -> A e. C)
12		prss.2	. . . . 5 |- B e. V
13	12	pri2 1842	. . . 4 |- B e. {A, B}
14		ssel 1502	. . . 4 |- ({A, B} (_ C -> (B e. {A, B} -> B e. C))
15	13, 14	mpi 44	. . 3 |- ({A, B} (_ C -> B e. C)
16	11, 15	jca 236	. 2 |- ({A, B} (_ C -> (A e. C /\ B e. C))
17	7, 16	impbi 139	1 |- ((A e. C /\ B e. C) <-> {A, B} (_ C)

Syntax hints:   <-> wb 127   \/ wo 195   /\ wa 196   = wceq 1091   e. wcel 1092  Vcvv 1348   (_ wss 1487  {cpr 1809

This theorem is referenced by:  pwssun 1917  fr2nr 2177  xpex 2488  fiint 3445  shincl 5332  chincl 5382

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491  df-ss 1492  df-sn 1811  df-pr 1812

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