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Theorem List for Intuitionistic Logic Explorer - 3201-3300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremss2abdv 3201* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} ⊆ {𝑥𝜒})

Theoremabssdv 3202* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
(𝜑 → (𝜓𝑥𝐴))       (𝜑 → {𝑥𝜓} ⊆ 𝐴)

Theoremabssi 3203* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
(𝜑𝑥𝐴)       {𝑥𝜑} ⊆ 𝐴

Theoremss2rab 3204 Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)
({𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓} ↔ ∀𝑥𝐴 (𝜑𝜓))

Theoremrabss 3205* Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
({𝑥𝐴𝜑} ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝜑𝑥𝐵))

Theoremssrab 3206* Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
(𝐵 ⊆ {𝑥𝐴𝜑} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))

Theoremssrabdv 3207* Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 31-Aug-2006.)
(𝜑𝐵𝐴)    &   ((𝜑𝑥𝐵) → 𝜓)       (𝜑𝐵 ⊆ {𝑥𝐴𝜓})

Theoremrabssdv 3208* Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 2-Feb-2015.)
((𝜑𝑥𝐴𝜓) → 𝑥𝐵)       (𝜑 → {𝑥𝐴𝜓} ⊆ 𝐵)

Theoremss2rabdv 3209* Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})

Theoremss2rabi 3210 Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)
(𝑥𝐴 → (𝜑𝜓))       {𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓}

Theoremrabss2 3211* Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 → {𝑥𝐴𝜑} ⊆ {𝑥𝐵𝜑})

Theoremssab2 3212* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)
{𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐴

Theoremssrab2 3213* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)
{𝑥𝐴𝜑} ⊆ 𝐴

Theoremssrabeq 3214* If the restricting class of a restricted class abstraction is a subset of this restricted class abstraction, it is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
(𝑉 ⊆ {𝑥𝑉𝜑} ↔ 𝑉 = {𝑥𝑉𝜑})

Theoremrabssab 3215 A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)
{𝑥𝐴𝜑} ⊆ {𝑥𝜑}

Theoremuniiunlem 3216* A subset relationship useful for converting union to indexed union using dfiun2 or dfiun2g and intersection to indexed intersection using dfiin2 . (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
(∀𝑥𝐴 𝐵𝐷 → (∀𝑥𝐴 𝐵𝐶 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶))

2.1.13  The difference, union, and intersection of two classes

2.1.13.1  The difference of two classes

Theoremdfdif3 3217* Alternate definition of class difference. Definition of relative set complement in Section 2.3 of [Pierik], p. 10. (Contributed by BJ and Jim Kingdon, 16-Jun-2022.)
(𝐴𝐵) = {𝑥𝐴 ∣ ∀𝑦𝐵 𝑥𝑦}

Theoremdifeq1 3218 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremdifeq2 3219 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremdifeq12 3220 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Theoremdifeq1i 3221 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)

Theoremdifeq2i 3222 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)

Theoremdifeq12i 3223 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)

Theoremdifeq1d 3224 Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Theoremdifeq2d 3225 Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Theoremdifeq12d 3226 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Theoremdifeqri 3227* Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ 𝑥𝐶)       (𝐴𝐵) = 𝐶

Theoremnfdif 3228 Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)

Theoremeldifi 3229 Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
(𝐴 ∈ (𝐵𝐶) → 𝐴𝐵)

Theoremeldifn 3230 Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)
(𝐴 ∈ (𝐵𝐶) → ¬ 𝐴𝐶)

Theoremelndif 3231 A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
(𝐴𝐵 → ¬ 𝐴 ∈ (𝐶𝐵))

Theoremdifdif 3232 Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
(𝐴 ∖ (𝐵𝐴)) = 𝐴

Theoremdifss 3233 Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
(𝐴𝐵) ⊆ 𝐴

Theoremdifssd 3234 A difference of two classes is contained in the minuend. Deduction form of difss 3233. (Contributed by David Moews, 1-May-2017.)
(𝜑 → (𝐴𝐵) ⊆ 𝐴)

Theoremdifss2 3235 If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.)
(𝐴 ⊆ (𝐵𝐶) → 𝐴𝐵)

Theoremdifss2d 3236 If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 3235. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ⊆ (𝐵𝐶))       (𝜑𝐴𝐵)

Theoremssdifss 3237 Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)
(𝐴𝐵 → (𝐴𝐶) ⊆ 𝐵)

Theoremddifnel 3238* Double complement under universal class. The hypothesis corresponds to stability of membership in 𝐴, which is weaker than decidability (see dcstab 830). Actually, the conclusion is a characterization of stability of membership in a class (see ddifstab 3239) . Exercise 4.10(s) of [Mendelson] p. 231, but with an additional hypothesis. For a version without a hypothesis, but which only states that 𝐴 is a subset of V ∖ (V ∖ 𝐴), see ddifss 3345. (Contributed by Jim Kingdon, 21-Jul-2018.)
𝑥 ∈ (V ∖ 𝐴) → 𝑥𝐴)       (V ∖ (V ∖ 𝐴)) = 𝐴

Theoremddifstab 3239* A class is equal to its double complement if and only if it is stable (that is, membership in it is a stable property). (Contributed by BJ, 12-Dec-2021.)
((V ∖ (V ∖ 𝐴)) = 𝐴 ↔ ∀𝑥STAB 𝑥𝐴)

Theoremssconb 3240 Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
((𝐴𝐶𝐵𝐶) → (𝐴 ⊆ (𝐶𝐵) ↔ 𝐵 ⊆ (𝐶𝐴)))

Theoremsscon 3241 Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)
(𝐴𝐵 → (𝐶𝐵) ⊆ (𝐶𝐴))

Theoremssdif 3242 Difference law for subsets. (Contributed by NM, 28-May-1998.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremssdifd 3243 If 𝐴 is contained in 𝐵, then (𝐴𝐶) is contained in (𝐵𝐶). Deduction form of ssdif 3242. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremsscond 3244 If 𝐴 is contained in 𝐵, then (𝐶𝐵) is contained in (𝐶𝐴). Deduction form of sscon 3241. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 → (𝐶𝐵) ⊆ (𝐶𝐴))

Theoremssdifssd 3245 If 𝐴 is contained in 𝐵, then (𝐴𝐶) is also contained in 𝐵. Deduction form of ssdifss 3237. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 → (𝐴𝐶) ⊆ 𝐵)

Theoremssdif2d 3246 If 𝐴 is contained in 𝐵 and 𝐶 is contained in 𝐷, then (𝐴𝐷) is contained in (𝐵𝐶). Deduction form. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐶𝐷)       (𝜑 → (𝐴𝐷) ⊆ (𝐵𝐶))

Theoremraldifb 3247 Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
(∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥 ∈ (𝐴𝐵)𝜑)

2.1.13.2  The union of two classes

Theoremelun 3248 Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.)
(𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))

Theoremuneqri 3249* Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶)       (𝐴𝐵) = 𝐶

Theoremunidm 3250 Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐴) = 𝐴

Theoremuncom 3251 Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵) = (𝐵𝐴)

Theoremequncom 3252 If a class equals the union of two other classes, then it equals the union of those two classes commuted. (Contributed by Alan Sare, 18-Feb-2012.)
(𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵))

Theoremequncomi 3253 Inference form of equncom 3252. (Contributed by Alan Sare, 18-Feb-2012.)
𝐴 = (𝐵𝐶)       𝐴 = (𝐶𝐵)

Theoremuneq1 3254 Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremuneq2 3255 Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremuneq12 3256 Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Theoremuneq1i 3257 Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)

Theoremuneq2i 3258 Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)

Theoremuneq12i 3259 Equality inference for union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)

Theoremuneq1d 3260 Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Theoremuneq2d 3261 Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Theoremuneq12d 3262 Equality deduction for union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Theoremnfun 3263 Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)

Theoremunass 3264 Associative law for union of classes. Exercise 8 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵𝐶))

Theoremun12 3265 A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
(𝐴 ∪ (𝐵𝐶)) = (𝐵 ∪ (𝐴𝐶))

Theoremun23 3266 A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ 𝐵)

Theoremun4 3267 A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)
((𝐴𝐵) ∪ (𝐶𝐷)) = ((𝐴𝐶) ∪ (𝐵𝐷))

Theoremunundi 3268 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Theoremunundir 3269 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Theoremssun1 3270 Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
𝐴 ⊆ (𝐴𝐵)

Theoremssun2 3271 Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.)
𝐴 ⊆ (𝐵𝐴)

Theoremssun3 3272 Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵𝐴 ⊆ (𝐵𝐶))

Theoremssun4 3273 Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
(𝐴𝐵𝐴 ⊆ (𝐶𝐵))

Theoremelun1 3274 Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵𝐴 ∈ (𝐵𝐶))

Theoremelun2 3275 Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
(𝐴𝐵𝐴 ∈ (𝐶𝐵))

Theoremunss1 3276 Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremssequn1 3277 A relationship between subclass and union. Theorem 26 of [Suppes] p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)

Theoremunss2 3278 Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))

Theoremunss12 3279 Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ⊆ (𝐵𝐷))

Theoremssequn2 3280 A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)
(𝐴𝐵 ↔ (𝐵𝐴) = 𝐵)

Theoremunss 3281 The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)
((𝐴𝐶𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)

Theoremunssi 3282 An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)
𝐴𝐶    &   𝐵𝐶       (𝐴𝐵) ⊆ 𝐶

Theoremunssd 3283 A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐶)    &   (𝜑𝐵𝐶)       (𝜑 → (𝐴𝐵) ⊆ 𝐶)

Theoremunssad 3284 If (𝐴𝐵) is contained in 𝐶, so is 𝐴. One-way deduction form of unss 3281. Partial converse of unssd 3283. (Contributed by David Moews, 1-May-2017.)
(𝜑 → (𝐴𝐵) ⊆ 𝐶)       (𝜑𝐴𝐶)

Theoremunssbd 3285 If (𝐴𝐵) is contained in 𝐶, so is 𝐵. One-way deduction form of unss 3281. Partial converse of unssd 3283. (Contributed by David Moews, 1-May-2017.)
(𝜑 → (𝐴𝐵) ⊆ 𝐶)       (𝜑𝐵𝐶)

Theoremssun 3286 A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)
((𝐴𝐵𝐴𝐶) → 𝐴 ⊆ (𝐵𝐶))

Theoremrexun 3287 Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)
(∃𝑥 ∈ (𝐴𝐵)𝜑 ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐵 𝜑))

Theoremralunb 3288 Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(∀𝑥 ∈ (𝐴𝐵)𝜑 ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜑))

Theoremralun 3289 Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)
((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜑) → ∀𝑥 ∈ (𝐴𝐵)𝜑)

2.1.13.3  The intersection of two classes

Theoremelin 3290 Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.)
(𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))

Theoremelini 3291 Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐴𝐵    &   𝐴𝐶       𝐴 ∈ (𝐵𝐶)

Theoremelind 3292 Deduce membership in an intersection of two classes. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝑋𝐴)    &   (𝜑𝑋𝐵)       (𝜑𝑋 ∈ (𝐴𝐵))

Theoremelinel1 3293 Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ (𝐵𝐶) → 𝐴𝐵)

Theoremelinel2 3294 Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ (𝐵𝐶) → 𝐴𝐶)

Theoremelin2 3295 Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑋 = (𝐵𝐶)       (𝐴𝑋 ↔ (𝐴𝐵𝐴𝐶))

Theoremelin1d 3296 Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)
(𝜑𝑋 ∈ (𝐴𝐵))       (𝜑𝑋𝐴)

Theoremelin2d 3297 Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)
(𝜑𝑋 ∈ (𝐴𝐵))       (𝜑𝑋𝐵)

Theoremelin3 3298 Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑋 = ((𝐵𝐶) ∩ 𝐷)       (𝐴𝑋 ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))

Theoremincom 3299 Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵) = (𝐵𝐴)

Theoremineqri 3300* Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.)
((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶)       (𝐴𝐵) = 𝐶

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