Theorem prssg 2476

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

Assertion

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

Proof of Theorem prssg

Step	Hyp	Ref	Expression
1		eleq1 1537	. . . 4 |- (x = A -> (x e. C <-> A e. C))
2	1	anbi1d 619	. . 3 |- (x = A -> ((x e. C /\ y e. C) <-> (A e. C /\ y e. C)))
3		preq1 2452	. . . 4 |- (x = A -> {x, y} = {A, y})
4	3	sseq1d 2091	. . 3 |- (x = A -> ({x, y} (_ C <-> {A, y} (_ C))
5	2, 4	bibi12d 631	. 2 |- (x = A -> (((x e. C /\ y e. C) <-> {x, y} (_ C) <-> ((A e. C /\ y e. C) <-> {A, y} (_ C)))
6		eleq1 1537	. . . 4 |- (y = B -> (y e. C <-> B e. C))
7	6	anbi2d 618	. . 3 |- (y = B -> ((A e. C /\ y e. C) <-> (A e. C /\ B e. C)))
8		preq2 2453	. . . 4 |- (y = B -> {A, y} = {A, B})
9	8	sseq1d 2091	. . 3 |- (y = B -> ({A, y} (_ C <-> {A, B} (_ C))
10	7, 9	bibi12d 631	. 2 |- (y = B -> (((A e. C /\ y e. C) <-> {A, y} (_ C) <-> ((A e. C /\ B e. C) <-> {A, B} (_ C)))
11		visset 1816	. . 3 |- x e. V
12		visset 1816	. . 3 |- y e. V
13	11, 12	prss 2475	. 2 |- ((x e. C /\ y e. C) <-> {x, y} (_ C)
14	5, 10, 13	vtocl2g 1853	1 |- ((A e. R /\ B e. S) -> ((A e. C /\ B e. C) <-> {A, B} (_ C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960   (_ wss 2050  {cpr 2414

This theorem is referenced by:  sspr 2479  set2elt 10531  cnfilca 10562

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-in 2054  df-ss 2056  df-sn 2416  df-pr 2417

Copyright terms: Public domain