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**Statement List for Metamath Proof Explorer -** 3101-3200 - Page 32 of 175
Type	Label	Description
Statement

Theorem	tpeq2 3101	Equality theorem for unordered triples.


Theorem	tpeq3 3102	Equality theorem for unordered triples.


Theorem	tprot 3103	Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)


Theorem	prid1g 3104	An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)


Theorem	prid2g 3105	An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)


Theorem	prid1 3106	An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49.


Theorem	prid2 3107	An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49.


Theorem	prprc1 3108	A proper class vanishes in an unordered pair.


Theorem	prprc2 3109	A proper class vanishes in an unordered pair.


Theorem	prprc 3110	An unordered pair containing two proper classes is the empty set.
$/\$

Theorem	tpid1 3111	One of the three elements of an unordered triple. (The proof was shortened by Andrew Salmon, 29-Jun-2011.)


Theorem	tpid1OLD 3112	One of the three elements of an unordered triple.


Theorem	tpid2 3113	One of the three elements of an unordered triple. (The proof was shortened by Andrew Salmon, 29-Jun-2011.)


Theorem	tpid2OLD 3114	One of the three elements of an unordered triple.


Theorem	tpid3g 3115	Closed theorem form of tpid3 3116. This proof was automatically generated from the virtual deduction proof tpid3gVD 16666 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)


Theorem	tpid3 3116	One of the three elements of an unordered triple. (The proof was shortened by Andrew Salmon, 29-Jun-2011.)


Theorem	tpid3OLD 3117	One of the three elements of an unordered triple.


Theorem	snnzg 3118	The singleton of a set is not empty.


Theorem	snnz 3119	The singleton of a set is not empty.


Theorem	prnz 3120	A pair containing a set is not empty.


Theorem	tpnz 3121	A triplet containing a set is not empty.


Theorem	snss 3122	The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49.


Theorem	eldifsn 3123	Membership in a set with an element removed.
$/\$

Theorem	snssg 3124	The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49.


Theorem	difsn 3125	An element not in a set can be removed without affecting the set. (The proof was shortened by Andrew Salmon, 29-Jun-2011.)


Theorem	difsnOLD 3126	An element not in a set can be removed without affecting the set.


Theorem	difprsn 3127	Removal of a singleton from an unordered pair. (The proof was shortened by Andrew Salmon, 29-Jun-2011.)


Theorem	difprsnOLD 3128	Removal of a singleton from an unordered pair.


Theorem	snssi 3129	The singleton of an element of a class is a subset of the class.


Theorem	snssd 3130	First-order logic and set theory. (Contributed by Jonathan Ben-Naim 12-Jun-2011).


Theorem	difsnid 3131	If we remove a single element from a class then put it back in, we end up with the original class.


Theorem	pw0 3132	Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47. (The proof was shortened by Andrew Salmon, 29-Jun-2011.)


Theorem	pw0OLD 3133	Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47. (The proof was shortened by Eric Schmidt, 4-Apr-2007.)


Theorem	pwpw0 3134	Compute the power set of the power set of the empty set. (See pw0 3132 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 3172, we have chosen to show a direct elementary proof.


Theorem	snsspr1 3135	A singleton is a subset of an unordered pair containing its member. (The proof was shortened by Andrew Salmon, 29-Jun-2011.)


Theorem	snsspr1OLD 3136	A singleton is a subset of an unordered pair containing its member.


Theorem	snsspr2 3137	A singleton is a subset of an unordered pair containing its member.


Theorem	prss 3138	A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (The proof was shortened by Andrew Salmon, 29-Jun-2011.)
$/\$

Theorem	prssOLD 3139	A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49.
$/\$

Theorem	prssg 3140	A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (The proof was shortened by Andrew Salmon, 29-Jun-2011.)
$/\$ $/\$

Theorem	prssgOLD 3141	A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49.
$/\$ $/\$

Theorem	sssn 3142	The subsets of a singleton.
$\/$

Theorem	eqsn 3143	Two ways to express that a nonempty set equals a singleton.


Theorem	sspr 3144	The subsets of a pair.
$\/$ $\/$ $\/$

Theorem	tpss 3145	A triplet of elements of a class is a subset of the class. (The proof was shortened by Andrew Salmon, 29-Jun-2011.)
$/\$ $/\$

Theorem	tpssOLD 3146	A triplet of elements of a class is a subset of the class.
$/\$ $/\$

Theorem	sneqr 3147	If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of [TakeutiZaring] p. 15.


Theorem	snsssn 3148	If a singleton is a subset of another, their members are equal.


Theorem	snsspw 3149	The singleton of a class is a subset of its power class.


Theorem	prsspw 3150	An unordered pair belongs to the power class of a class iff each member belongs to the class. (The proof was shortened by Andrew Salmon, 26-Jun-2011.)
$/\$

Theorem	prsspwOLD 3151	An unordered pair belongs to the power class of a class iff each member belongs to the class.
$/\$

Theorem	preqr1 3152	Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal.


Theorem	preqr2 3153	Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal.


Theorem	preq12b 3154	Equality relationship for two unordered pairs.
$/\$ $\/$ $/\$

Theorem	prel12 3155	Equality of two unordered pairs.
$/\$

Theorem	opthpr 3156	A way to represent ordered pairs using unordered pairs with distinct members.
$/\$

Theorem	preqsn 3157	Equivalence for a pair equal to a singleton.
$/\$

Theorem	opeq1 3158	Equality theorem for ordered pairs.


Theorem	opeq2 3159	Equality theorem for ordered pairs.


Theorem	opeq12 3160	Equality theorem for ordered pairs.
$/\$

Theorem	opeq1i 3161	Equality inference for ordered pairs.


Theorem	opeq2i 3162	Equality inference for ordered pairs.


Theorem	opeq12i 3163	Equality inference for ordered pairs. (The proof was shortened by Eric Schmidt, 4-Apr-2007.)


Theorem	opeq1d 3164	Equality deduction for ordered pairs.


Theorem	opeq2d 3165	Equality deduction for ordered pairs.


Theorem	opeq12d 3166	Equality deduction for ordered pairs. (The proof was shortened by Andrew Salmon, 29-Jun-2011.)


Theorem	opeq12dOLD 3167	Equality deduction for ordered pairs.


Theorem	hbop 3168	Bound-variable hypothesis builder for ordered pairs.


Theorem	hbopd 3169	Deduction version of bound-variable hypothesis builder hbop 3168.


Theorem	opprc1 3170	Expansion of an ordered pair when the first member is a proper class. See also opprc1b 3542, opprc2 3171, opprc3 3543.


Theorem	opprc2 3171	A property of an ordered pair of proper classes (due to our particular definition of ordered pair).


Theorem	pwsn 3172	The power set of a singleton.


Theorem	pwsnALT 3173	The power set of a singleton (direct proof).


Theorem	pwpr 3174	The power set of an unordered pair.


Theorem	pwpwpw0 3175	Compute the power set of the power set of the power set of the empty set. (See also pw0 3132 and pwpw0 3134.)


Theorem	pwv 3176	The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235.


The union of a class

Syntax	cuni 3177	Extend class notation to include the union of a class (read: 'union ')


Definition	df-uni 3178	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.
$/\$

Theorem	dfuni2 3179	Alternate definition of class union.


Theorem	eluni 3180	Membership in class union.
$/\$

Theorem	eluni2 3181	Membership in class union. Restricted quantifier version.


Theorem	elunii 3182	Membership in class union.
$/\$

Theorem	hbuni 3183	Bound-variable hypothesis builder for union. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)


Theorem	hbuniOLD 3184	Bound-variable hypothesis builder for union.


Theorem	unieq 3185	Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18. (The proof was shortened by Andrew Salmon, 29-Jun-2011.)


Theorem	unieqOLD 3186	Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18.


Theorem	unieqi 3187	Inference of equality of two class unions.


Theorem	unieqd 3188	Deduction of equality of two class unions.


Theorem	eluniab 3189	Membership in union of a class abstraction.
$/\$

Theorem	elunirab 3190	Membership in union of a class abstraction.
$/\$

Theorem	unipr 3191	The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16.


Theorem	uniprg 3192	The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16.
$/\$

Theorem	unisn 3193	A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.


Theorem	unisng 3194	A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.


Theorem	hbeqel 3195	If is effectively bound in , then it is effectively bound in . This is the other direction of hbeleq 1997, showing the two conditions are equivalent when is a set.


Theorem	uniun 3196	The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53.


Theorem	uniin 3197	The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uninqs 14340 for a condition where equality holds. (The proof was shortened by Andrew Salmon, 29-Jun-2011.)


Theorem	uniinOLD 3198	The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uninqs 14340 for a condition where equality holds.


Theorem	uniss 3199	Subclass relationship for class union. Theorem 61 of [Suppes] p. 39. (The proof was shortened by Andrew Salmon, 29-Jun-2011.)


Theorem	unissOLD 3200	Subclass relationship for class union. Theorem 61 of [Suppes] p. 39.

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