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Theorem List for Intuitionistic Logic Explorer - 3601-3700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempwsnss 3601 The power set of a singleton. (Contributed by Jim Kingdon, 12-Aug-2018.)
{∅, {𝐴}} ⊆ 𝒫 {𝐴}

Theorempwpw0ss 3602 Compute the power set of the power set of the empty set. (See pw0 3538 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.)
{∅, {∅}} ⊆ 𝒫 {∅}

Theorempwprss 3603 The power set of an unordered pair. (Contributed by Jim Kingdon, 13-Aug-2018.)
({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ⊆ 𝒫 {𝐴, 𝐵}

Theorempwtpss 3604 The power set of an unordered triple. (Contributed by Jim Kingdon, 13-Aug-2018.)
(({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}})) ⊆ 𝒫 {𝐴, 𝐵, 𝐶}

Theorempwpwpw0ss 3605 Compute the power set of the power set of the power set of the empty set. (See also pw0 3538 and pwpw0ss 3602.) (Contributed by Jim Kingdon, 13-Aug-2018.)
({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ⊆ 𝒫 {∅, {∅}}

Theorempwv 3606 The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)
𝒫 V = V

2.1.18  The union of a class

Syntaxcuni 3607 Extend class notation to include the union of a class (read: 'union 𝐴')
class 𝐴

Definitiondf-uni 3608* 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 2949. (Contributed by NM, 23-Aug-1993.)
𝐴 = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦𝐴)}

Theoremdfuni2 3609* Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)
𝐴 = {𝑥 ∣ ∃𝑦𝐴 𝑥𝑦}

Theoremeluni 3610* Membership in class union. (Contributed by NM, 22-May-1994.)
(𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))

Theoremeluni2 3611* Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)
(𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)

Theoremelunii 3612 Membership in class union. (Contributed by NM, 24-Mar-1995.)
((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)

Theoremnfuni 3613 Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝑥𝐴       𝑥 𝐴

Theoremnfunid 3614 Deduction version of nfuni 3613. (Contributed by NM, 18-Feb-2013.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)

Theoremcsbunig 3615 Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.)
(𝐴𝑉𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)

Theoremunieq 3616 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.)
(𝐴 = 𝐵 𝐴 = 𝐵)

Theoremunieqi 3617 Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵        𝐴 = 𝐵

Theoremunieqd 3618 Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 𝐴 = 𝐵)

Theoremeluniab 3619* Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)
(𝐴 {𝑥𝜑} ↔ ∃𝑥(𝐴𝑥𝜑))

Theoremelunirab 3620* Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)
(𝐴 {𝑥𝐵𝜑} ↔ ∃𝑥𝐵 (𝐴𝑥𝜑))

Theoremunipr 3621 The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.)
𝐴 ∈ V    &   𝐵 ∈ V        {𝐴, 𝐵} = (𝐴𝐵)

Theoremuniprg 3622 The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))

Theoremunisn 3623 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 30-Aug-1993.)
𝐴 ∈ V        {𝐴} = 𝐴

Theoremunisng 3624 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)
(𝐴𝑉 {𝐴} = 𝐴)

Theoremdfnfc2 3625* An alternative statement of the effective freeness of a class 𝐴, when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.)
(∀𝑥 𝐴𝑉 → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))

Theoremuniun 3626 The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53. (Contributed by NM, 20-Aug-1993.)
(𝐴𝐵) = ( 𝐴 𝐵)

Theoremuniin 3627 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.)
(𝐴𝐵) ⊆ ( 𝐴 𝐵)

Theoremuniss 3628 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.)
(𝐴𝐵 𝐴 𝐵)

Theoremssuni 3629 Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)

Theoremunissi 3630 Subclass relationship for subclass union. Inference form of uniss 3628. (Contributed by David Moews, 1-May-2017.)
𝐴𝐵        𝐴 𝐵

Theoremunissd 3631 Subclass relationship for subclass union. Deduction form of uniss 3628. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 𝐴 𝐵)

Theoremuni0b 3632 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.)
( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})

Theoremuni0c 3633* The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)
( 𝐴 = ∅ ↔ ∀𝑥𝐴 𝑥 = ∅)

Theoremuni0 3634 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.)
∅ = ∅

Theoremelssuni 3635 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.)
(𝐴𝐵𝐴 𝐵)

Theoremunissel 3636 Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.)
(( 𝐴𝐵𝐵𝐴) → 𝐴 = 𝐵)

Theoremunissb 3637* Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.)
( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)

Theoremuniss2 3638* 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.)
(∀𝑥𝐴𝑦𝐵 𝑥𝑦 𝐴 𝐵)

Theoremunidif 3639* If the difference 𝐴𝐵 contains the largest members of 𝐴, then the union of the difference is the union of 𝐴. (Contributed by NM, 22-Mar-2004.)
(∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 (𝐴𝐵) = 𝐴)

Theoremssunieq 3640* Relationship implying union. (Contributed by NM, 10-Nov-1999.)
((𝐴𝐵 ∧ ∀𝑥𝐵 𝑥𝐴) → 𝐴 = 𝐵)

Theoremunimax 3641* Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.)
(𝐴𝐵 {𝑥𝐵𝑥𝐴} = 𝐴)

2.1.19  The intersection of a class

Syntaxcint 3642 Extend class notation to include the intersection of a class (read: 'intersect 𝐴').
class 𝐴

Definitiondf-int 3643* 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 2951. (Contributed by NM, 18-Aug-1993.)
𝐴 = {𝑥 ∣ ∀𝑦(𝑦𝐴𝑥𝑦)}

Theoremdfint2 3644* Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)
𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}

Theoreminteq 3645 Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
(𝐴 = 𝐵 𝐴 = 𝐵)

Theoreminteqi 3646 Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)
𝐴 = 𝐵        𝐴 = 𝐵

Theoreminteqd 3647 Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)
(𝜑𝐴 = 𝐵)       (𝜑 𝐴 = 𝐵)

Theoremelint 3648* Membership in class intersection. (Contributed by NM, 21-May-1994.)
𝐴 ∈ V       (𝐴 𝐵 ↔ ∀𝑥(𝑥𝐵𝐴𝑥))

Theoremelint2 3649* Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
𝐴 ∈ V       (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)

Theoremelintg 3650* Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)
(𝐴𝑉 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))

Theoremelinti 3651 Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝐴 𝐵 → (𝐶𝐵𝐴𝐶))

Theoremnfint 3652 Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
𝑥𝐴       𝑥 𝐴

Theoremelintab 3653* Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)
𝐴 ∈ V       (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))

Theoremelintrab 3654* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)
𝐴 ∈ V       (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥))

Theoremelintrabg 3655* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)
(𝐴𝑉 → (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥)))

Theoremint0 3656 The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)
∅ = V

Theoremintss1 3657 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.)
(𝐴𝐵 𝐵𝐴)

Theoremssint 3658* Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)
(𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)

Theoremssintab 3659* Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))

Theoremssintub 3660* Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
𝐴 {𝑥𝐵𝐴𝑥}

Theoremssmin 3661* Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
𝐴 {𝑥 ∣ (𝐴𝑥𝜑)}

Theoremintmin 3662* 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.)
(𝐴𝐵 {𝑥𝐵𝐴𝑥} = 𝐴)

Theoremintss 3663 Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
(𝐴𝐵 𝐵 𝐴)

Theoremintssunim 3664* The intersection of an inhabited set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
(∃𝑥 𝑥𝐴 𝐴 𝐴)

Theoremssintrab 3665* Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)
(𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥))

Theoremintssuni2m 3666* Subclass relationship for intersection and union. (Contributed by Jim Kingdon, 14-Aug-2018.)
((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴) → 𝐴 𝐵)

Theoremintminss 3667* Under subset ordering, the intersection of a restricted class abstraction is less than or equal to any of its members. (Contributed by NM, 7-Sep-2013.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓) → {𝑥𝐵𝜑} ⊆ 𝐴)

Theoremintmin2 3668* Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)
𝐴 ∈ V        {𝑥𝐴𝑥} = 𝐴

Theoremintmin3 3669* Under subset ordering, the intersection of a class abstraction is less than or equal to any of its members. (Contributed by NM, 3-Jul-2005.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝜓       (𝐴𝑉 {𝑥𝜑} ⊆ 𝐴)

Theoremintmin4 3670* Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)
(𝐴 {𝑥𝜑} → {𝑥 ∣ (𝐴𝑥𝜑)} = {𝑥𝜑})

Theoremintab 3671* The intersection of a special case of a class abstraction. 𝑦 may be free in 𝜑 and 𝐴, which can be thought of a 𝜑(𝑦) and 𝐴(𝑦). (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
𝐴 ∈ V    &   {𝑥 ∣ ∃𝑦(𝜑𝑥 = 𝐴)} ∈ V        {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} = {𝑥 ∣ ∃𝑦(𝜑𝑥 = 𝐴)}

Theoremint0el 3672 The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
(∅ ∈ 𝐴 𝐴 = ∅)

Theoremintun 3673 The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42. (Contributed by NM, 22-Sep-2002.)
(𝐴𝐵) = ( 𝐴 𝐵)

Theoremintpr 3674 The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.)
𝐴 ∈ V    &   𝐵 ∈ V        {𝐴, 𝐵} = (𝐴𝐵)

Theoremintprg 3675 The intersection of a pair is the intersection of its members. Closed form of intpr 3674. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))

Theoremintsng 3676 Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐴𝑉 {𝐴} = 𝐴)

Theoremintsn 3677 The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41. (Contributed by NM, 29-Sep-2002.)
𝐴 ∈ V        {𝐴} = 𝐴

Theoremuniintsnr 3678* The union and intersection of a singleton are equal. See also eusn 3471. (Contributed by Jim Kingdon, 14-Aug-2018.)
(∃𝑥 𝐴 = {𝑥} → 𝐴 = 𝐴)

Theoremuniintabim 3679 The union and the intersection of a class abstraction are equal if there is a unique satisfying value of 𝜑(𝑥). (Contributed by Jim Kingdon, 14-Aug-2018.)
(∃!𝑥𝜑 {𝑥𝜑} = {𝑥𝜑})

Theoremintunsn 3680 Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)
𝐵 ∈ V        (𝐴 ∪ {𝐵}) = ( 𝐴𝐵)

Theoremrint0 3681 Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 = ∅ → (𝐴 𝑋) = 𝐴)

Theoremelrint 3682* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 ∈ (𝐴 𝐵) ↔ (𝑋𝐴 ∧ ∀𝑦𝐵 𝑋𝑦))

Theoremelrint2 3683* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋𝐴 → (𝑋 ∈ (𝐴 𝐵) ↔ ∀𝑦𝐵 𝑋𝑦))

2.1.20  Indexed union and intersection

Syntaxciun 3684 Extend class notation to include indexed union. Note: Historically (prior to 21-Oct-2005), set.mm used the notation 𝑥𝐴𝐵, with the same union symbol as cuni 3607. While that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses as distinguished symbol instead of and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.
class 𝑥𝐴 𝐵

Syntaxciin 3685 Extend class notation to include indexed intersection. Note: Historically (prior to 21-Oct-2005), set.mm used the notation 𝑥𝐴𝐵, with the same intersection symbol as cint 3642. Although that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses a distinguished symbol instead of and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.
class 𝑥𝐴 𝐵

Definitiondf-iun 3686* Define indexed union. Definition indexed union in [Stoll] p. 45. In most applications, 𝐴 is independent of 𝑥 (although this is not required by the definition), and 𝐵 depends on 𝑥 i.e. can be read informally as 𝐵(𝑥). We call 𝑥 the index, 𝐴 the index set, and 𝐵 the indexed set. In most books, 𝑥𝐴 is written as a subscript or underneath a union symbol . We use a special union symbol to make it easier to distinguish from plain class union. In many theorems, you will see that 𝑥 and 𝐴 are in the same distinct variable group (meaning 𝐴 cannot depend on 𝑥) and that 𝐵 and 𝑥 do not share a distinct variable group (meaning that can be thought of as 𝐵(𝑥) i.e. can be substituted with a class expression containing 𝑥). An alternate definition tying indexed union to ordinary union is dfiun2 3718. Theorem uniiun 3737 provides a definition of ordinary union in terms of indexed union. (Contributed by NM, 27-Jun-1998.)
𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}

Definitiondf-iin 3687* Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3686. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 3719. Theorem intiin 3738 provides a definition of ordinary intersection in terms of indexed intersection. (Contributed by NM, 27-Jun-1998.)
𝑥𝐴 𝐵 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}

Theoremeliun 3688* Membership in indexed union. (Contributed by NM, 3-Sep-2003.)
(𝐴 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝐴𝐶)

Theoremeliin 3689* Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)
(𝐴𝑉 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))

Theoremiuncom 3690* Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)
𝑥𝐴 𝑦𝐵 𝐶 = 𝑦𝐵 𝑥𝐴 𝐶

Theoremiuncom4 3691 Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.)
𝑥𝐴 𝐵 = 𝑥𝐴 𝐵

Theoremiunconstm 3692* Indexed union of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by Jim Kingdon, 15-Aug-2018.)
(∃𝑥 𝑥𝐴 𝑥𝐴 𝐵 = 𝐵)

Theoremiinconstm 3693* Indexed intersection of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by Jim Kingdon, 19-Dec-2018.)
(∃𝑦 𝑦𝐴 𝑥𝐴 𝐵 = 𝐵)

Theoremiuniin 3694* Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝑥𝐴 𝑦𝐵 𝐶 𝑦𝐵 𝑥𝐴 𝐶

Theoremiunss1 3695* Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴𝐵 𝑥𝐴 𝐶 𝑥𝐵 𝐶)

Theoremiinss1 3696* Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)
(𝐴𝐵 𝑥𝐵 𝐶 𝑥𝐴 𝐶)

Theoremiuneq1 3697* Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)
(𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Theoremiineq1 3698* Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)
(𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Theoremss2iun 3699 Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 𝑥𝐴 𝐶)

Theoremiuneq2 3700 Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)
(∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

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