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Theorem List for Metamath Proof Explorer - 3501-3600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsseq12d 3501 An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))
 
Theoremeqsstri 3502 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
𝐴 = 𝐵    &   𝐵𝐶       𝐴𝐶
 
Theoremeqsstr3i 3503 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
𝐵 = 𝐴    &   𝐵𝐶       𝐴𝐶
 
Theoremsseqtri 3504 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
𝐴𝐵    &   𝐵 = 𝐶       𝐴𝐶
 
Theoremsseqtr4i 3505 Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)
𝐴𝐵    &   𝐶 = 𝐵       𝐴𝐶
 
Theoremeqsstrd 3506 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremeqsstr3d 3507 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐵 = 𝐴)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsseqtrd 3508 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)
 
Theoremsseqtr4d 3509 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)
 
Theorem3sstr3i 3510 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
𝐴𝐵    &   𝐴 = 𝐶    &   𝐵 = 𝐷       𝐶𝐷
 
Theorem3sstr4i 3511 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
𝐴𝐵    &   𝐶 = 𝐴    &   𝐷 = 𝐵       𝐶𝐷
 
Theorem3sstr3g 3512 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
(𝜑𝐴𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝐷)
 
Theorem3sstr4g 3513 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝐷)
 
Theorem3sstr3d 3514 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝐷)
 
Theorem3sstr4d 3515 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝐷)
 
Theoremsyl5eqss 3516 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
𝐴 = 𝐵    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsyl5eqssr 3517 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
𝐵 = 𝐴    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsyl6sseq 3518 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴𝐵)    &   𝐵 = 𝐶       (𝜑𝐴𝐶)
 
Theoremsyl6sseqr 3519 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐵       (𝜑𝐴𝐶)
 
Theoremsyl5sseq 3520 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
𝐵𝐴    &   (𝜑𝐴 = 𝐶)       (𝜑𝐵𝐶)
 
Theoremsyl5sseqr 3521 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
𝐵𝐴    &   (𝜑𝐶 = 𝐴)       (𝜑𝐵𝐶)
 
Theoremsyl6eqss 3522 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   𝐵𝐶       (𝜑𝐴𝐶)
 
Theoremsyl6eqssr 3523 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝐵 = 𝐴)    &   𝐵𝐶       (𝜑𝐴𝐶)
 
Theoremeqimss 3524 Equality implies the subclass relation. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(𝐴 = 𝐵𝐴𝐵)
 
Theoremeqimss2 3525 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)
(𝐵 = 𝐴𝐴𝐵)
 
Theoremeqimssi 3526 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)
𝐴 = 𝐵       𝐴𝐵
 
Theoremeqimss2i 3527 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)
𝐴 = 𝐵       𝐵𝐴
 
Theoremnssne1 3528 Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)
((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)
 
Theoremnssne2 3529 Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)
((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)
 
Theoremnss 3530* Negation of subclass relationship. Exercise 13 of [TakeutiZaring] p. 18. (Contributed by NM, 25-Feb-1996.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
𝐴𝐵 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
 
Theoremnelss 3531 Demonstrate by witnesses that two classes lack a subclass relation. (Contributed by Stefan O'Rear, 5-Feb-2015.)
((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵𝐶)
 
Theoremssrexf 3532 restricted existential quantification follows from a subclass relationship. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐴    &   𝑥𝐵       (𝐴𝐵 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜑))
 
Theoremssralv 3533* Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)
(𝐴𝐵 → (∀𝑥𝐵 𝜑 → ∀𝑥𝐴 𝜑))
 
Theoremssrexv 3534* Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)
(𝐴𝐵 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜑))
 
Theoremralss 3535* Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝐴𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 (𝑥𝐴𝜑)))
 
Theoremrexss 3536* Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝐴𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 (𝑥𝐴𝜑)))
 
Theoremss2ab 3537 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
({𝑥𝜑} ⊆ {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))
 
Theoremabss 3538* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
({𝑥𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
 
Theoremssab 3539* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
(𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
 
Theoremssabral 3540* The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)
(𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥𝐴 𝜑)
 
Theoremss2abi 3541 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)
(𝜑𝜓)       {𝑥𝜑} ⊆ {𝑥𝜓}
 
Theoremss2abdv 3542* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} ⊆ {𝑥𝜒})
 
Theoremabssdv 3543* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
(𝜑 → (𝜓𝑥𝐴))       (𝜑 → {𝑥𝜓} ⊆ 𝐴)
 
Theoremabssi 3544* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
(𝜑𝑥𝐴)       {𝑥𝜑} ⊆ 𝐴
 
Theoremss2rab 3545 Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)
({𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓} ↔ ∀𝑥𝐴 (𝜑𝜓))
 
Theoremrabss 3546* Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
({𝑥𝐴𝜑} ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝜑𝑥𝐵))
 
Theoremssrab 3547* Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
(𝐵 ⊆ {𝑥𝐴𝜑} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))
 
Theoremssrabdv 3548* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 31-Aug-2006.)
(𝜑𝐵𝐴)    &   ((𝜑𝑥𝐵) → 𝜓)       (𝜑𝐵 ⊆ {𝑥𝐴𝜓})
 
Theoremrabssdv 3549* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 2-Feb-2015.)
((𝜑𝑥𝐴𝜓) → 𝑥𝐵)       (𝜑 → {𝑥𝐴𝜓} ⊆ 𝐵)
 
Theoremss2rabdv 3550* Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})
 
Theoremss2rabi 3551 Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)
(𝑥𝐴 → (𝜑𝜓))       {𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓}
 
Theoremrabss2 3552* Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 → {𝑥𝐴𝜑} ⊆ {𝑥𝐵𝜑})
 
Theoremssab2 3553* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)
{𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
 
Theoremssrab2 3554* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)
{𝑥𝐴𝜑} ⊆ 𝐴
 
Theoremssrabeq 3555* 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 3556 A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)
{𝑥𝐴𝜑} ⊆ {𝑥𝜑}
 
Theoremuniiunlem 3557* A subset relationship useful for converting union to indexed union using dfiun2 4388 or dfiun2g 4386 and intersection to indexed intersection using dfiin2 4389. (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
(∀𝑥𝐴 𝐵𝐷 → (∀𝑥𝐴 𝐵𝐶 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶))
 
Theoremdfpss2 3558 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)
(𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
 
Theoremdfpss3 3559 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
 
Theorempsseq1 3560 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
 
Theorempsseq2 3561 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
 
Theorempsseq1i 3562 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)
 
Theorempsseq2i 3563 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
𝐴 = 𝐵       (𝐶𝐴𝐶𝐵)
 
Theorempsseq12i 3564 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶𝐵𝐷)
 
Theorempsseq1d 3565 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))
 
Theorempsseq2d 3566 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))
 
Theorempsseq12d 3567 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))
 
Theorempssss 3568 A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
(𝐴𝐵𝐴𝐵)
 
Theorempssne 3569 Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.)
(𝐴𝐵𝐴𝐵)
 
Theorempssssd 3570 Deduce subclass from proper subclass. (Contributed by NM, 29-Feb-1996.)
(𝜑𝐴𝐵)       (𝜑𝐴𝐵)
 
Theorempssned 3571 Proper subclasses are unequal. Deduction form of pssne 3569. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑𝐴𝐵)
 
Theoremsspss 3572 Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.)
(𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
 
Theorempssirr 3573 Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
¬ 𝐴𝐴
 
Theorempssn2lp 3574 Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
¬ (𝐴𝐵𝐵𝐴)
 
Theoremsspsstri 3575 Two ways of stating trichotomy with respect to inclusion. (Contributed by NM, 12-Aug-2004.)
((𝐴𝐵𝐵𝐴) ↔ (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
 
Theoremssnpss 3576 Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 → ¬ 𝐵𝐴)
 
Theorempsstr 3577 Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremsspsstr 3578 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theorempsssstr 3579 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theorempsstrd 3580 Proper subclass inclusion is transitive. Deduction form of psstr 3577. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsspsstrd 3581 Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3578. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theorempsssstrd 3582 Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 3579. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremnpss 3583 A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3487. (Contributed by Mario Carneiro, 15-May-2015.)
𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
 
Theoremssnelpss 3584 A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)
(𝐴𝐵 → ((𝐶𝐵 ∧ ¬ 𝐶𝐴) → 𝐴𝐵))
 
Theoremssnelpssd 3585 Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3584. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐶𝐵)    &   (𝜑 → ¬ 𝐶𝐴)       (𝜑𝐴𝐵)
 
Theoremssexnelpss 3586* If there is an element of a class which is not contained in a subclass, the subclass is a proper subclass. (Contributed by AV, 29-Jan-2020.)
((𝐴𝐵 ∧ ∃𝑥𝐵 𝑥𝐴) → 𝐴𝐵)
 
2.1.13  The difference, union, and intersection of two classes
 
2.1.13.1  The difference of two classes
 
Theoremdifeq1 3587 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremdifeq2 3588 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremdifeq12 3589 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
 
Theoremdifeq1i 3590 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)
 
Theoremdifeq2i 3591 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)
 
Theoremdifeq12i 3592 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)
 
Theoremdifeq1d 3593 Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremdifeq2d 3594 Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremdifeq12d 3595 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))
 
Theoremdifeqri 3596* 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 3597 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 3598 Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
(𝐴 ∈ (𝐵𝐶) → 𝐴𝐵)
 
Theoremeldifn 3599 Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)
(𝐴 ∈ (𝐵𝐶) → ¬ 𝐴𝐶)
 
Theoremelndif 3600 A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
(𝐴𝐵 → ¬ 𝐴 ∈ (𝐶𝐵))
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