Theorem prss 2468

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 1541	. . . . 5 |- (A e. C -> (x = A -> x e. C))
2		eleq1a 1541	. . . . 5 |- (B e. C -> (x = B -> x e. C))
3	1, 2	jaao 427	. . . 4 |- ((A e. C /\ B e. C) -> ((x = A \/ x = B) -> x e. C))
4		visset 1810	. . . . 5 |- x e. V
5	4	elpr 2421	. . . 4 |- (x e. {A, B} <-> (x = A \/ x = B))
6	3, 5	syl5ib 206	. . 3 |- ((A e. C /\ B e. C) -> (x e. {A, B} -> x e. C))
7	6	ssrdv 2067	. 2 |- ((A e. C /\ B e. C) -> {A, B} (_ C)
8		prss.1	. . . . 5 |- A e. V
9	8	pri1 2447	. . . 4 |- A e. {A, B}
10		ssel 2060	. . . 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 2448	. . . 4 |- B e. {A, B}
14		ssel 2060	. . . 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 288	. 2 |- ({A, B} (_ C -> (A e. C /\ B e. C))
17	7, 16	impbi 157	1 |- ((A e. C /\ B e. C) <-> {A, B} (_ C)

Syntax hints:   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 955   e. wcel 957  Vcvv 1808   (_ wss 2044  {cpr 2407

This theorem is referenced by:  prssg 2469  pwssun 2823  fr2nr 2921  xpsspw 3253  fiint 4543  rankelun 4690  shincl 9286  chincl 9338  clicls 10538

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809  df-un 2047  df-in 2048  df-ss 2050  df-sn 2409  df-pr 2410

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