Theorem List for Intuitionistic Logic Explorer - 3701-3800 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | preqr1g 3701 |
Reverse equality lemma for unordered pairs. If two unordered pairs have
the same second element, the first elements are equal. Closed form of
preqr1 3703. (Contributed by Jim Kingdon, 21-Sep-2018.)
|
![( (](lp.gif) ![( (](lp.gif) ![_V _V](rmcv.gif) ![( (](lp.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![C C](_cc.gif) ![{ {](lbrace.gif) ![B B](_cb.gif)
![C C](_cc.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | preqr2g 3702 |
Reverse equality lemma for unordered pairs. If two unordered pairs have
the same second element, the second elements are equal. Closed form of
preqr2 3704. (Contributed by Jim Kingdon, 21-Sep-2018.)
|
![( (](lp.gif) ![( (](lp.gif) ![_V _V](rmcv.gif) ![( (](lp.gif) ![{ {](lbrace.gif) ![C C](_cc.gif) ![A A](_ca.gif) ![{ {](lbrace.gif) ![C C](_cc.gif)
![B B](_cb.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | preqr1 3703 |
Reverse equality lemma for unordered pairs. If two unordered pairs have
the same second element, the first elements are equal. (Contributed by
NM, 18-Oct-1995.)
|
![( (](lp.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![C C](_cc.gif)
![{ {](lbrace.gif) ![B B](_cb.gif)
![C C](_cc.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | preqr2 3704 |
Reverse equality lemma for unordered pairs. If two unordered pairs have
the same first element, the second elements are equal. (Contributed by
NM, 5-Aug-1993.)
|
![( (](lp.gif) ![{ {](lbrace.gif) ![C C](_cc.gif) ![A A](_ca.gif)
![{ {](lbrace.gif) ![C C](_cc.gif)
![B B](_cb.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | preq12b 3705 |
Equality relationship for two unordered pairs. (Contributed by NM,
17-Oct-1996.)
|
![( (](lp.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![B B](_cb.gif)
![{ {](lbrace.gif) ![C C](_cc.gif)
![D D](_cd.gif) ![( (](lp.gif) ![( (](lp.gif) ![D D](_cd.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | prel12 3706 |
Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.)
|
![( (](lp.gif) ![( (](lp.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![{ {](lbrace.gif) ![C C](_cc.gif)
![D D](_cd.gif) ![( (](lp.gif) ![{ {](lbrace.gif) ![C C](_cc.gif) ![D D](_cd.gif) ![{ {](lbrace.gif) ![C C](_cc.gif) ![D D](_cd.gif) ![} }](rbrace.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | opthpr 3707 |
A way to represent ordered pairs using unordered pairs with distinct
members. (Contributed by NM, 27-Mar-2007.)
|
![( (](lp.gif) ![( (](lp.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![B B](_cb.gif)
![{ {](lbrace.gif) ![C C](_cc.gif)
![D D](_cd.gif) ![( (](lp.gif)
![D D](_cd.gif) ![) )](rp.gif) ![)
)](rp.gif) ![) )](rp.gif) |
|
Theorem | preq12bg 3708 |
Closed form of preq12b 3705. (Contributed by Scott Fenton,
28-Mar-2014.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![W W](_cw.gif) ![( (](lp.gif)
![Y Y](_cy.gif) ![) )](rp.gif)
![( (](lp.gif) ![{ {](lbrace.gif) ![A A](_ca.gif)
![B B](_cb.gif) ![{ {](lbrace.gif) ![C C](_cc.gif) ![D D](_cd.gif) ![( (](lp.gif) ![( (](lp.gif) ![D D](_cd.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | prneimg 3709 |
Two pairs are not equal if at least one element of the first pair is not
contained in the second pair. (Contributed by Alexander van der Vekens,
13-Aug-2017.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![V V](_cv.gif) ![( (](lp.gif)
![Y Y](_cy.gif) ![) )](rp.gif)
![( (](lp.gif) ![( (](lp.gif) ![(
(](lp.gif) ![D D](_cd.gif) ![( (](lp.gif) ![D D](_cd.gif) ![) )](rp.gif)
![{ {](lbrace.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![{ {](lbrace.gif) ![C C](_cc.gif)
![D D](_cd.gif) ![} }](rbrace.gif) ![)
)](rp.gif) ![) )](rp.gif) |
|
Theorem | preqsn 3710 |
Equivalence for a pair equal to a singleton. (Contributed by NM,
3-Jun-2008.)
|
![( (](lp.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![{ {](lbrace.gif) ![C C](_cc.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | dfopg 3711 |
Value of the ordered pair when the arguments are sets. (Contributed by
Mario Carneiro, 26-Apr-2015.)
|
![( (](lp.gif) ![( (](lp.gif) ![W W](_cw.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![{ {](lbrace.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![} }](rbrace.gif)
![{ {](lbrace.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![} }](rbrace.gif) ![} }](rbrace.gif) ![) )](rp.gif) |
|
Theorem | dfop 3712 |
Value of an ordered pair when the arguments are sets, with the
conclusion corresponding to Kuratowski's original definition.
(Contributed by NM, 25-Jun-1998.)
|
![<.
<.](langle.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![{ {](lbrace.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![} }](rbrace.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![} }](rbrace.gif) ![} }](rbrace.gif) |
|
Theorem | opeq1 3713 |
Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.)
(Revised by Mario Carneiro, 26-Apr-2015.)
|
![( (](lp.gif) ![<.
<.](langle.gif) ![A A](_ca.gif) ![C C](_cc.gif) ![<.
<.](langle.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![>. >.](rangle.gif) ![) )](rp.gif) |
|
Theorem | opeq2 3714 |
Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.)
(Revised by Mario Carneiro, 26-Apr-2015.)
|
![( (](lp.gif) ![<.
<.](langle.gif) ![C C](_cc.gif) ![A A](_ca.gif) ![<.
<.](langle.gif) ![C C](_cc.gif) ![B B](_cb.gif) ![>. >.](rangle.gif) ![) )](rp.gif) |
|
Theorem | opeq12 3715 |
Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
|
![( (](lp.gif) ![( (](lp.gif) ![D D](_cd.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![<. <.](langle.gif) ![C C](_cc.gif) ![D D](_cd.gif) ![>. >.](rangle.gif) ![) )](rp.gif) |
|
Theorem | opeq1i 3716 |
Equality inference for ordered pairs. (Contributed by NM,
16-Dec-2006.)
|
![<. <.](langle.gif) ![A A](_ca.gif) ![C C](_cc.gif) ![<. <.](langle.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![>. >.](rangle.gif) |
|
Theorem | opeq2i 3717 |
Equality inference for ordered pairs. (Contributed by NM,
16-Dec-2006.)
|
![<. <.](langle.gif) ![C C](_cc.gif) ![A A](_ca.gif) ![<. <.](langle.gif) ![C C](_cc.gif) ![B B](_cb.gif) ![>. >.](rangle.gif) |
|
Theorem | opeq12i 3718 |
Equality inference for ordered pairs. (Contributed by NM,
16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
|
![<.
<.](langle.gif) ![A A](_ca.gif) ![C C](_cc.gif) ![<.
<.](langle.gif) ![B B](_cb.gif) ![D D](_cd.gif) ![>. >.](rangle.gif) |
|
Theorem | opeq1d 3719 |
Equality deduction for ordered pairs. (Contributed by NM,
16-Dec-2006.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![C C](_cc.gif) ![<. <.](langle.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![>. >.](rangle.gif) ![) )](rp.gif) |
|
Theorem | opeq2d 3720 |
Equality deduction for ordered pairs. (Contributed by NM,
16-Dec-2006.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![<. <.](langle.gif) ![C C](_cc.gif) ![A A](_ca.gif) ![<. <.](langle.gif) ![C C](_cc.gif) ![B B](_cb.gif) ![>. >.](rangle.gif) ![) )](rp.gif) |
|
Theorem | opeq12d 3721 |
Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
(Proof shortened by Andrew Salmon, 29-Jun-2011.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![D D](_cd.gif) ![( (](lp.gif) ![<. <.](langle.gif) ![A A](_ca.gif)
![C C](_cc.gif) ![<.
<.](langle.gif) ![B B](_cb.gif) ![D D](_cd.gif) ![>. >.](rangle.gif) ![) )](rp.gif) |
|
Theorem | oteq1 3722 |
Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
|
![( (](lp.gif) ![<.
<.](langle.gif) ![A A](_ca.gif) ![C C](_cc.gif) ![D D](_cd.gif) ![<. <.](langle.gif) ![B B](_cb.gif) ![C C](_cc.gif)
![D D](_cd.gif) ![>. >.](rangle.gif) ![) )](rp.gif) |
|
Theorem | oteq2 3723 |
Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
|
![( (](lp.gif) ![<.
<.](langle.gif) ![C C](_cc.gif) ![A A](_ca.gif) ![D D](_cd.gif) ![<. <.](langle.gif) ![C C](_cc.gif) ![B B](_cb.gif)
![D D](_cd.gif) ![>. >.](rangle.gif) ![) )](rp.gif) |
|
Theorem | oteq3 3724 |
Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
|
![( (](lp.gif) ![<.
<.](langle.gif) ![C C](_cc.gif) ![D D](_cd.gif) ![A A](_ca.gif) ![<. <.](langle.gif) ![C C](_cc.gif) ![D D](_cd.gif)
![B B](_cb.gif) ![>. >.](rangle.gif) ![) )](rp.gif) |
|
Theorem | oteq1d 3725 |
Equality deduction for ordered triples. (Contributed by Mario Carneiro,
11-Jan-2017.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![C C](_cc.gif)
![D D](_cd.gif) ![<.
<.](langle.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![D D](_cd.gif) ![>. >.](rangle.gif) ![) )](rp.gif) |
|
Theorem | oteq2d 3726 |
Equality deduction for ordered triples. (Contributed by Mario Carneiro,
11-Jan-2017.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![<. <.](langle.gif) ![C C](_cc.gif) ![A A](_ca.gif)
![D D](_cd.gif) ![<.
<.](langle.gif) ![C C](_cc.gif) ![B B](_cb.gif) ![D D](_cd.gif) ![>. >.](rangle.gif) ![) )](rp.gif) |
|
Theorem | oteq3d 3727 |
Equality deduction for ordered triples. (Contributed by Mario Carneiro,
11-Jan-2017.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![<. <.](langle.gif) ![C C](_cc.gif) ![D D](_cd.gif)
![A A](_ca.gif) ![<.
<.](langle.gif) ![C C](_cc.gif) ![D D](_cd.gif) ![B B](_cb.gif) ![>. >.](rangle.gif) ![) )](rp.gif) |
|
Theorem | oteq123d 3728 |
Equality deduction for ordered triples. (Contributed by Mario Carneiro,
11-Jan-2017.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![D D](_cd.gif) ![( (](lp.gif) ![F F](_cf.gif) ![( (](lp.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![C C](_cc.gif)
![E E](_ce.gif) ![<.
<.](langle.gif) ![B B](_cb.gif) ![D D](_cd.gif) ![F F](_cf.gif) ![>. >.](rangle.gif) ![) )](rp.gif) |
|
Theorem | nfop 3729 |
Bound-variable hypothesis builder for ordered pairs. (Contributed by
NM, 14-Nov-1995.)
|
![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![>. >.](rangle.gif) |
|
Theorem | nfopd 3730 |
Deduction version of bound-variable hypothesis builder nfop 3729.
This
shows how the deduction version of a not-free theorem such as nfop 3729
can
be created from the corresponding not-free inference theorem.
(Contributed by NM, 4-Feb-2008.)
|
![( (](lp.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![A A](_ca.gif) ![( (](lp.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![B B](_cb.gif) ![( (](lp.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![>. >.](rangle.gif) ![) )](rp.gif) |
|
Theorem | opid 3731 |
The ordered pair ![<. <.](langle.gif) ![A A](_ca.gif) ![A A](_ca.gif) in Kuratowski's representation.
(Contributed by FL, 28-Dec-2011.)
|
![<. <.](langle.gif) ![A A](_ca.gif) ![A A](_ca.gif) ![{ {](lbrace.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![} }](rbrace.gif) ![} }](rbrace.gif) |
|
Theorem | ralunsn 3732* |
Restricted quantification over the union of a set and a singleton, using
implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)
(Revised by Mario Carneiro, 23-Apr-2015.)
|
![( (](lp.gif) ![( (](lp.gif) ![ps ps](_psi.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![A. A.](forall.gif) ![( (](lp.gif) ![{ {](lbrace.gif) ![B B](_cb.gif) ![} }](rbrace.gif) ![) )](rp.gif) ![( (](lp.gif) ![A. A.](forall.gif)
![ps ps](_psi.gif) ![) )](rp.gif) ![)
)](rp.gif) ![) )](rp.gif) |
|
Theorem | 2ralunsn 3733* |
Double restricted quantification over the union of a set and a
singleton, using implicit substitution. (Contributed by Paul Chapman,
17-Nov-2012.)
|
![( (](lp.gif) ![( (](lp.gif) ![ch ch](_chi.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![ps ps](_psi.gif) ![) )](rp.gif) ![( (](lp.gif)
![( (](lp.gif) ![th th](_theta.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![A. A.](forall.gif) ![( (](lp.gif) ![{ {](lbrace.gif) ![B B](_cb.gif) ![} }](rbrace.gif) ![) )](rp.gif) ![A. A.](forall.gif) ![( (](lp.gif) ![{ {](lbrace.gif) ![B B](_cb.gif) ![} }](rbrace.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![A. A.](forall.gif)
![A. A.](forall.gif)
![A. A.](forall.gif) ![ps ps](_psi.gif) ![( (](lp.gif) ![A. A.](forall.gif) ![th th](_theta.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | opprc 3734 |
Expansion of an ordered pair when either member is a proper class.
(Contributed by Mario Carneiro, 26-Apr-2015.)
|
![( (](lp.gif) ![( (](lp.gif) ![_V _V](rmcv.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![(/) (/)](varnothing.gif) ![) )](rp.gif) |
|
Theorem | opprc1 3735 |
Expansion of an ordered pair when the first member is a proper class. See
also opprc 3734. (Contributed by NM, 10-Apr-2004.) (Revised
by Mario
Carneiro, 26-Apr-2015.)
|
![( (](lp.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![(/) (/)](varnothing.gif) ![) )](rp.gif) |
|
Theorem | opprc2 3736 |
Expansion of an ordered pair when the second member is a proper class.
See also opprc 3734. (Contributed by NM, 15-Nov-1994.) (Revised
by Mario
Carneiro, 26-Apr-2015.)
|
![( (](lp.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![(/) (/)](varnothing.gif) ![) )](rp.gif) |
|
Theorem | oprcl 3737 |
If an ordered pair has an element, then its arguments are sets.
(Contributed by Mario Carneiro, 26-Apr-2015.)
|
![( (](lp.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![( (](lp.gif)
![_V _V](rmcv.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | pwsnss 3738 |
The power set of a singleton. (Contributed by Jim Kingdon,
12-Aug-2018.)
|
![{
{](lbrace.gif) ![(/) (/)](varnothing.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![} }](rbrace.gif)
![~P ~P](scrp.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![} }](rbrace.gif) |
|
Theorem | pwpw0ss 3739 |
Compute the power set of the power set of the empty set. (See pw0 3675
for
the power set of the empty set.) Theorem 90 of [Suppes] p. 48 (but with
subset in place of equality). (Contributed by Jim Kingdon,
12-Aug-2018.)
|
![{
{](lbrace.gif) ![(/) (/)](varnothing.gif) ![{ {](lbrace.gif) ![(/) (/)](varnothing.gif) ![} }](rbrace.gif) ![~P
~P](scrp.gif) ![{ {](lbrace.gif) ![(/) (/)](varnothing.gif) ![} }](rbrace.gif) |
|
Theorem | pwprss 3740 |
The power set of an unordered pair. (Contributed by Jim Kingdon,
13-Aug-2018.)
|
![( (](lp.gif) ![{ {](lbrace.gif) ![(/) (/)](varnothing.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![} }](rbrace.gif) ![{ {](lbrace.gif) ![{ {](lbrace.gif) ![B B](_cb.gif) ![} }](rbrace.gif) ![{ {](lbrace.gif) ![A A](_ca.gif)
![B B](_cb.gif) ![} }](rbrace.gif) ![} }](rbrace.gif)
![~P ~P](scrp.gif) ![{ {](lbrace.gif) ![A A](_ca.gif)
![B B](_cb.gif) ![} }](rbrace.gif) |
|
Theorem | pwtpss 3741 |
The power set of an unordered triple. (Contributed by Jim Kingdon,
13-Aug-2018.)
|
![( (](lp.gif) ![( (](lp.gif) ![{ {](lbrace.gif) ![(/) (/)](varnothing.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![} }](rbrace.gif) ![{ {](lbrace.gif) ![{ {](lbrace.gif) ![B B](_cb.gif) ![} }](rbrace.gif) ![{ {](lbrace.gif) ![A A](_ca.gif)
![B B](_cb.gif) ![} }](rbrace.gif) ![} }](rbrace.gif)
![( (](lp.gif) ![{ {](lbrace.gif) ![{ {](lbrace.gif) ![C C](_cc.gif) ![} }](rbrace.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![C C](_cc.gif) ![} }](rbrace.gif) ![{ {](lbrace.gif) ![{ {](lbrace.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![} }](rbrace.gif) ![{ {](lbrace.gif) ![A A](_ca.gif)
![B B](_cb.gif) ![C C](_cc.gif) ![} }](rbrace.gif) ![} }](rbrace.gif) ![) )](rp.gif) ![~P ~P](scrp.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![B B](_cb.gif)
![C C](_cc.gif) ![} }](rbrace.gif) |
|
Theorem | pwpwpw0ss 3742 |
Compute the power set of the power set of the power set of the empty set.
(See also pw0 3675 and pwpw0ss 3739.) (Contributed by Jim Kingdon,
13-Aug-2018.)
|
![( (](lp.gif) ![{ {](lbrace.gif) ![(/) (/)](varnothing.gif) ![{ {](lbrace.gif) ![(/) (/)](varnothing.gif) ![}
}](rbrace.gif) ![{ {](lbrace.gif) ![{ {](lbrace.gif) ![{ {](lbrace.gif) ![(/) (/)](varnothing.gif) ![} }](rbrace.gif) ![} }](rbrace.gif) ![{ {](lbrace.gif) ![(/) (/)](varnothing.gif) ![{ {](lbrace.gif) ![(/) (/)](varnothing.gif) ![} }](rbrace.gif) ![} }](rbrace.gif) ![} }](rbrace.gif)
![~P ~P](scrp.gif) ![{ {](lbrace.gif) ![(/) (/)](varnothing.gif) ![{ {](lbrace.gif) ![(/) (/)](varnothing.gif) ![} }](rbrace.gif) ![} }](rbrace.gif) |
|
Theorem | pwv 3743 |
The power class of the universe is the universe. Exercise 4.12(d) of
[Mendelson] p. 235. (Contributed by NM,
14-Sep-2003.)
|
![~P
~P](scrp.gif)
![_V _V](rmcv.gif) |
|
2.1.18 The union of a class
|
|
Syntax | cuni 3744 |
Extend class notation to include the union of a class (read: 'union
')
|
![U. U.](bigcup.gif) ![A A](_ca.gif) |
|
Definition | df-uni 3745* |
Define the union of a class i.e. the collection of all members of the
members of the class. Definition 5.5 of [TakeutiZaring] p. 16. For
example, { { 1 , 3 } , { 1 , 8 } } = { 1 , 3 , 8 } . This is similar to
the union of two classes df-un 3080. (Contributed by NM, 23-Aug-1993.)
|
![U.
U.](bigcup.gif) ![{ {](lbrace.gif)
![E. E.](exists.gif) ![y y](_y.gif) ![(
(](lp.gif)
![A A](_ca.gif) ![) )](rp.gif) ![} }](rbrace.gif) |
|
Theorem | dfuni2 3746* |
Alternate definition of class union. (Contributed by NM,
28-Jun-1998.)
|
![U.
U.](bigcup.gif) ![{ {](lbrace.gif)
![E. E.](exists.gif) ![y y](_y.gif) ![} }](rbrace.gif) |
|
Theorem | eluni 3747* |
Membership in class union. (Contributed by NM, 22-May-1994.)
|
![( (](lp.gif) ![U. U.](bigcup.gif) ![E.
E.](exists.gif) ![x x](_x.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | eluni2 3748* |
Membership in class union. Restricted quantifier version. (Contributed
by NM, 31-Aug-1999.)
|
![( (](lp.gif) ![U. U.](bigcup.gif) ![E.
E.](exists.gif)
![x x](_x.gif) ![) )](rp.gif) |
|
Theorem | elunii 3749 |
Membership in class union. (Contributed by NM, 24-Mar-1995.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![U. U.](bigcup.gif) ![C C](_cc.gif) ![) )](rp.gif) |
|
Theorem | nfuni 3750 |
Bound-variable hypothesis builder for union. (Contributed by NM,
30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|
![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![U. U.](bigcup.gif) ![A A](_ca.gif) |
|
Theorem | nfunid 3751 |
Deduction version of nfuni 3750. (Contributed by NM, 18-Feb-2013.)
|
![( (](lp.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![A A](_ca.gif) ![( (](lp.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![U. U.](bigcup.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | csbunig 3752 |
Distribute proper substitution through the union of a class.
(Contributed by Alan Sare, 10-Nov-2012.)
|
![( (](lp.gif) ![[_
[_](_ulbrack.gif) ![x x](_x.gif) ![]_ ]_](_urbrack.gif) ![U.
U.](bigcup.gif) ![U. U.](bigcup.gif) ![[_ [_](_ulbrack.gif) ![x x](_x.gif) ![]_ ]_](_urbrack.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | unieq 3753 |
Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18.
(Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon,
29-Jun-2011.)
|
![( (](lp.gif) ![U. U.](bigcup.gif) ![U. U.](bigcup.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | unieqi 3754 |
Inference of equality of two class unions. (Contributed by NM,
30-Aug-1993.)
|
![U. U.](bigcup.gif) ![U. U.](bigcup.gif) ![B B](_cb.gif) |
|
Theorem | unieqd 3755 |
Deduction of equality of two class unions. (Contributed by NM,
21-Apr-1995.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![U. U.](bigcup.gif) ![U. U.](bigcup.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | eluniab 3756* |
Membership in union of a class abstraction. (Contributed by NM,
11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)
|
![( (](lp.gif) ![U. U.](bigcup.gif) ![{ {](lbrace.gif) ![ph ph](_varphi.gif) ![E. E.](exists.gif) ![x x](_x.gif) ![( (](lp.gif) ![ph ph](_varphi.gif) ![) )](rp.gif) ![)
)](rp.gif) |
|
Theorem | elunirab 3757* |
Membership in union of a class abstraction. (Contributed by NM,
4-Oct-2006.)
|
![( (](lp.gif) ![U. U.](bigcup.gif) ![{ {](lbrace.gif) ![ph ph](_varphi.gif) ![E.
E.](exists.gif) ![( (](lp.gif)
![ph ph](_varphi.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | unipr 3758 |
The union of a pair is the union of its members. Proposition 5.7 of
[TakeutiZaring] p. 16.
(Contributed by NM, 23-Aug-1993.)
|
![U.
U.](bigcup.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![B B](_cb.gif)
![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | uniprg 3759 |
The union of a pair is the union of its members. Proposition 5.7 of
[TakeutiZaring] p. 16.
(Contributed by NM, 25-Aug-2006.)
|
![( (](lp.gif) ![( (](lp.gif) ![W W](_cw.gif) ![U. U.](bigcup.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![B B](_cb.gif)
![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | unisn 3760 |
A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
(Contributed by NM, 30-Aug-1993.)
|
![U. U.](bigcup.gif) ![{ {](lbrace.gif) ![A A](_ca.gif)
![A A](_ca.gif) |
|
Theorem | unisng 3761 |
A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
(Contributed by NM, 13-Aug-2002.)
|
![( (](lp.gif) ![U. U.](bigcup.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | dfnfc2 3762* |
An alternate statement of the effective freeness of a class , when
it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.)
|
![( (](lp.gif) ![A. A.](forall.gif)
![( (](lp.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif)
![A. A.](forall.gif) ![y y](_y.gif) ![F/ F/](finv.gif)
![A A](_ca.gif) ![) )](rp.gif) ![)
)](rp.gif) |
|
Theorem | uniun 3763 |
The class union of the union of two classes. Theorem 8.3 of [Quine]
p. 53. (Contributed by NM, 20-Aug-1993.)
|
![U.
U.](bigcup.gif) ![( (](lp.gif) ![B B](_cb.gif)
![( (](lp.gif) ![U. U.](bigcup.gif)
![U. U.](bigcup.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | uniin 3764 |
The class union of the intersection of two classes. Exercise 4.12(n) of
[Mendelson] p. 235. (Contributed by
NM, 4-Dec-2003.) (Proof shortened
by Andrew Salmon, 29-Jun-2011.)
|
![U.
U.](bigcup.gif) ![( (](lp.gif) ![B B](_cb.gif)
![( (](lp.gif) ![U. U.](bigcup.gif) ![U. U.](bigcup.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | uniss 3765 |
Subclass relationship for class union. Theorem 61 of [Suppes] p. 39.
(Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon,
29-Jun-2011.)
|
![( (](lp.gif) ![U. U.](bigcup.gif) ![U. U.](bigcup.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | ssuni 3766 |
Subclass relationship for class union. (Contributed by NM,
24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|
![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![U. U.](bigcup.gif) ![C C](_cc.gif) ![) )](rp.gif) |
|
Theorem | unissi 3767 |
Subclass relationship for subclass union. Inference form of uniss 3765.
(Contributed by David Moews, 1-May-2017.)
|
![U.
U.](bigcup.gif) ![U. U.](bigcup.gif) ![B B](_cb.gif) |
|
Theorem | unissd 3768 |
Subclass relationship for subclass union. Deduction form of uniss 3765.
(Contributed by David Moews, 1-May-2017.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![U. U.](bigcup.gif) ![U.
U.](bigcup.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | uni0b 3769 |
The union of a set is empty iff the set is included in the singleton of
the empty set. (Contributed by NM, 12-Sep-2004.)
|
![( (](lp.gif) ![U. U.](bigcup.gif) ![{ {](lbrace.gif) ![(/) (/)](varnothing.gif) ![} }](rbrace.gif) ![) )](rp.gif) |
|
Theorem | uni0c 3770* |
The union of a set is empty iff all of its members are empty.
(Contributed by NM, 16-Aug-2006.)
|
![( (](lp.gif) ![U. U.](bigcup.gif) ![A. A.](forall.gif) ![(/) (/)](varnothing.gif) ![) )](rp.gif) |
|
Theorem | uni0 3771 |
The union of the empty set is the empty set. Theorem 8.7 of [Quine]
p. 54. (Reproved without relying on ax-nul by Eric Schmidt.)
(Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt,
4-Apr-2007.)
|
![U.
U.](bigcup.gif) ![(/) (/)](varnothing.gif) |
|
Theorem | elssuni 3772 |
An element of a class is a subclass of its union. Theorem 8.6 of [Quine]
p. 54. Also the basis for Proposition 7.20 of [TakeutiZaring] p. 40.
(Contributed by NM, 6-Jun-1994.)
|
![( (](lp.gif)
![U. U.](bigcup.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | unissel 3773 |
Condition turning a subclass relationship for union into an equality.
(Contributed by NM, 18-Jul-2006.)
|
![( (](lp.gif) ![( (](lp.gif) ![U. U.](bigcup.gif) ![A A](_ca.gif)
![U. U.](bigcup.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | unissb 3774* |
Relationship involving membership, subset, and union. Exercise 5 of
[Enderton] p. 26 and its converse.
(Contributed by NM, 20-Sep-2003.)
|
![( (](lp.gif) ![U. U.](bigcup.gif) ![A.
A.](forall.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | uniss2 3775* |
A subclass condition on the members of two classes that implies a
subclass relation on their unions. Proposition 8.6 of [TakeutiZaring]
p. 59. (Contributed by NM, 22-Mar-2004.)
|
![( (](lp.gif) ![A. A.](forall.gif) ![E. E.](exists.gif) ![U. U.](bigcup.gif) ![U. U.](bigcup.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | unidif 3776* |
If the difference
contains the largest
members of , then
the union of the difference is the union of . (Contributed by NM,
22-Mar-2004.)
|
![( (](lp.gif) ![A. A.](forall.gif) ![E. E.](exists.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif)
![U. U.](bigcup.gif) ![( (](lp.gif) ![B B](_cb.gif)
![U. U.](bigcup.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | ssunieq 3777* |
Relationship implying union. (Contributed by NM, 10-Nov-1999.)
|
![( (](lp.gif) ![( (](lp.gif) ![A. A.](forall.gif) ![A A](_ca.gif) ![U. U.](bigcup.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | unimax 3778* |
Any member of a class is the largest of those members that it includes.
(Contributed by NM, 13-Aug-2002.)
|
![( (](lp.gif) ![U. U.](bigcup.gif) ![{ {](lbrace.gif) ![A A](_ca.gif)
![A A](_ca.gif) ![) )](rp.gif) |
|
2.1.19 The intersection of a class
|
|
Syntax | cint 3779 |
Extend class notation to include the intersection of a class (read:
'intersect ').
|
![|^| |^|](bigcap.gif) ![A A](_ca.gif) |
|
Definition | df-int 3780* |
Define the intersection of a class. Definition 7.35 of [TakeutiZaring]
p. 44. For example, { { 1 , 3 } , { 1 , 8 } } = { 1 } .
Compare this with the intersection of two classes, df-in 3082.
(Contributed by NM, 18-Aug-1993.)
|
![|^|
|^|](bigcap.gif)
![{ {](lbrace.gif)
![A. A.](forall.gif) ![y y](_y.gif) ![( (](lp.gif) ![y y](_y.gif) ![) )](rp.gif) ![} }](rbrace.gif) |
|
Theorem | dfint2 3781* |
Alternate definition of class intersection. (Contributed by NM,
28-Jun-1998.)
|
![|^|
|^|](bigcap.gif)
![{ {](lbrace.gif)
![A. A.](forall.gif) ![y y](_y.gif) ![} }](rbrace.gif) |
|
Theorem | inteq 3782 |
Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
|
![( (](lp.gif) ![|^|
|^|](bigcap.gif)
![|^| |^|](bigcap.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | inteqi 3783 |
Equality inference for class intersection. (Contributed by NM,
2-Sep-2003.)
|
![|^| |^|](bigcap.gif) ![|^| |^|](bigcap.gif) ![B B](_cb.gif) |
|
Theorem | inteqd 3784 |
Equality deduction for class intersection. (Contributed by NM,
2-Sep-2003.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![|^| |^|](bigcap.gif) ![|^| |^|](bigcap.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | elint 3785* |
Membership in class intersection. (Contributed by NM, 21-May-1994.)
|
![( (](lp.gif) ![|^| |^|](bigcap.gif) ![A.
A.](forall.gif) ![x x](_x.gif) ![( (](lp.gif) ![x x](_x.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | elint2 3786* |
Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
|
![( (](lp.gif) ![|^| |^|](bigcap.gif) ![A.
A.](forall.gif)
![x x](_x.gif) ![) )](rp.gif) |
|
Theorem | elintg 3787* |
Membership in class intersection, with the sethood requirement expressed
as an antecedent. (Contributed by NM, 20-Nov-2003.)
|
![( (](lp.gif) ![( (](lp.gif) ![|^| |^|](bigcap.gif) ![A. A.](forall.gif)
![x x](_x.gif) ![) )](rp.gif) ![)
)](rp.gif) |
|
Theorem | elinti 3788 |
Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
(Proof shortened by Andrew Salmon, 9-Jul-2011.)
|
![( (](lp.gif) ![|^| |^|](bigcap.gif) ![( (](lp.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | nfint 3789 |
Bound-variable hypothesis builder for intersection. (Contributed by NM,
2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
|
![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![|^| |^|](bigcap.gif) ![A A](_ca.gif) |
|
Theorem | elintab 3790* |
Membership in the intersection of a class abstraction. (Contributed by
NM, 30-Aug-1993.)
|
![( (](lp.gif) ![|^| |^|](bigcap.gif) ![{
{](lbrace.gif) ![ph ph](_varphi.gif) ![A. A.](forall.gif) ![x x](_x.gif) ![( (](lp.gif) ![x x](_x.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | elintrab 3791* |
Membership in the intersection of a class abstraction. (Contributed by
NM, 17-Oct-1999.)
|
![( (](lp.gif) ![|^| |^|](bigcap.gif) ![{
{](lbrace.gif) ![ph ph](_varphi.gif) ![A. A.](forall.gif) ![( (](lp.gif)
![x x](_x.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | elintrabg 3792* |
Membership in the intersection of a class abstraction. (Contributed by
NM, 17-Feb-2007.)
|
![( (](lp.gif) ![( (](lp.gif) ![|^| |^|](bigcap.gif) ![{ {](lbrace.gif) ![ph ph](_varphi.gif) ![A.
A.](forall.gif) ![( (](lp.gif)
![x x](_x.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | int0 3793 |
The intersection of the empty set is the universal class. Exercise 2 of
[TakeutiZaring] p. 44.
(Contributed by NM, 18-Aug-1993.)
|
![|^|
|^|](bigcap.gif) ![_V _V](rmcv.gif) |
|
Theorem | intss1 3794 |
An element of a class includes the intersection of the class. Exercise
4 of [TakeutiZaring] p. 44 (with
correction), generalized to classes.
(Contributed by NM, 18-Nov-1995.)
|
![( (](lp.gif) ![|^|
|^|](bigcap.gif)
![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | ssint 3795* |
Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52
and its converse. (Contributed by NM, 14-Oct-1999.)
|
![( (](lp.gif) ![|^| |^|](bigcap.gif) ![A. A.](forall.gif) ![x x](_x.gif) ![) )](rp.gif) |
|
Theorem | ssintab 3796* |
Subclass of the intersection of a class abstraction. (Contributed by
NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|
![( (](lp.gif) ![|^| |^|](bigcap.gif) ![{ {](lbrace.gif) ![ph ph](_varphi.gif) ![A. A.](forall.gif) ![x x](_x.gif) ![( (](lp.gif)
![x x](_x.gif) ![) )](rp.gif) ![)
)](rp.gif) |
|
Theorem | ssintub 3797* |
Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
|
![|^| |^|](bigcap.gif) ![{ {](lbrace.gif) ![x x](_x.gif) ![} }](rbrace.gif) |
|
Theorem | ssmin 3798* |
Subclass of the minimum value of class of supersets. (Contributed by
NM, 10-Aug-2006.)
|
![|^| |^|](bigcap.gif) ![{ {](lbrace.gif) ![( (](lp.gif) ![ph ph](_varphi.gif) ![) )](rp.gif) ![} }](rbrace.gif) |
|
Theorem | intmin 3799* |
Any member of a class is the smallest of those members that include it.
(Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon,
9-Jul-2011.)
|
![( (](lp.gif) ![|^|
|^|](bigcap.gif) ![{ {](lbrace.gif)
![x x](_x.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | intss 3800 |
Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
|
![( (](lp.gif) ![|^|
|^|](bigcap.gif)
![|^| |^|](bigcap.gif) ![A A](_ca.gif) ![) )](rp.gif) |