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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | psseq12d 3701 | An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷)) | ||
Theorem | pssss 3702 | A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) |
⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) | ||
Theorem | pssne 3703 | Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.) |
⊢ (𝐴 ⊊ 𝐵 → 𝐴 ≠ 𝐵) | ||
Theorem | pssssd 3704 | Deduce subclass from proper subclass. (Contributed by NM, 29-Feb-1996.) |
⊢ (𝜑 → 𝐴 ⊊ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | ||
Theorem | pssned 3705 | Proper subclasses are unequal. Deduction form of pssne 3703. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊊ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) | ||
Theorem | sspss 3706 | Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.) |
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵)) | ||
Theorem | pssirr 3707 | Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) |
⊢ ¬ 𝐴 ⊊ 𝐴 | ||
Theorem | pssn2lp 3708 | 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.) |
⊢ ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐴) | ||
Theorem | sspsstri 3709 | Two ways of stating trichotomy with respect to inclusion. (Contributed by NM, 12-Aug-2004.) |
⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴)) | ||
Theorem | ssnpss 3710 | Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ⊊ 𝐴) | ||
Theorem | psstr 3711 | Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) |
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) | ||
Theorem | sspsstr 3712 | Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) | ||
Theorem | psssstr 3713 | Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.) |
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶) | ||
Theorem | psstrd 3714 | Proper subclass inclusion is transitive. Deduction form of psstr 3711. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊊ 𝐵) & ⊢ (𝜑 → 𝐵 ⊊ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊊ 𝐶) | ||
Theorem | sspsstrd 3715 | Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3712. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐵 ⊊ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊊ 𝐶) | ||
Theorem | psssstrd 3716 | Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 3713. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊊ 𝐵) & ⊢ (𝜑 → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊊ 𝐶) | ||
Theorem | npss 3717 | A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3618. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (¬ 𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵)) | ||
Theorem | ssnelpss 3718 | A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.) |
⊢ (𝐴 ⊆ 𝐵 → ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐴 ⊊ 𝐵)) | ||
Theorem | ssnelpssd 3719 | Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3718. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 ⊊ 𝐵) | ||
Theorem | ssexnelpss 3720* | 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.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝑥 ∉ 𝐴) → 𝐴 ⊊ 𝐵) | ||
Theorem | difeq1 3721 | Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)) | ||
Theorem | difeq2 3722 | Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 = 𝐵 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)) | ||
Theorem | difeq12 3723 | Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)) | ||
Theorem | difeq1i 3724 | Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶) | ||
Theorem | difeq2i 3725 | Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵) | ||
Theorem | difeq12i 3726 | Equality inference for class difference. (Contributed by NM, 29-Aug-2004.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷) | ||
Theorem | difeq1d 3727 | Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)) | ||
Theorem | difeq2d 3728 | Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)) | ||
Theorem | difeq12d 3729 | Equality deduction for class difference. (Contributed by FL, 29-May-2014.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)) | ||
Theorem | difeqri 3730* | Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ⇒ ⊢ (𝐴 ∖ 𝐵) = 𝐶 | ||
Theorem | nfdif 3731 | Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵) | ||
Theorem | eldifi 3732 | Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.) |
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → 𝐴 ∈ 𝐵) | ||
Theorem | eldifn 3733 | Implication of membership in a class difference. (Contributed by NM, 3-May-1994.) |
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → ¬ 𝐴 ∈ 𝐶) | ||
Theorem | elndif 3734 | A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.) |
⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵)) | ||
Theorem | neldif 3735 | Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994.) |
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ 𝐶)) → 𝐴 ∈ 𝐶) | ||
Theorem | difdif 3736 | Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.) |
⊢ (𝐴 ∖ (𝐵 ∖ 𝐴)) = 𝐴 | ||
Theorem | difss 3737 | Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.) |
⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | ||
Theorem | difssd 3738 | A difference of two classes is contained in the minuend. Deduction form of difss 3737. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴) | ||
Theorem | difss2 3739 | If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵) | ||
Theorem | difss2d 3740 | If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 3739. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∖ 𝐶)) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | ||
Theorem | ssdifss 3741 | Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ 𝐵) | ||
Theorem | ddif 3742 | Double complement under universal class. Exercise 4.10(s) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.) |
⊢ (V ∖ (V ∖ 𝐴)) = 𝐴 | ||
Theorem | ssconb 3743 | Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.) |
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ 𝐵 ⊆ (𝐶 ∖ 𝐴))) | ||
Theorem | sscon 3744 | Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) | ||
Theorem | ssdif 3745 | Difference law for subsets. (Contributed by NM, 28-May-1998.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) | ||
Theorem | ssdifd 3746 | If 𝐴 is contained in 𝐵, then (𝐴 ∖ 𝐶) is contained in (𝐵 ∖ 𝐶). Deduction form of ssdif 3745. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) | ||
Theorem | sscond 3747 | If 𝐴 is contained in 𝐵, then (𝐶 ∖ 𝐵) is contained in (𝐶 ∖ 𝐴). Deduction form of sscon 3744. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) | ||
Theorem | ssdifssd 3748 | If 𝐴 is contained in 𝐵, then (𝐴 ∖ 𝐶) is also contained in 𝐵. Deduction form of ssdifss 3741. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ 𝐵) | ||
Theorem | ssdif2d 3749 | If 𝐴 is contained in 𝐵 and 𝐶 is contained in 𝐷, then (𝐴 ∖ 𝐷) is contained in (𝐵 ∖ 𝐶). Deduction form. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐶 ⊆ 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐵 ∖ 𝐶)) | ||
Theorem | raldifb 3750 | Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.) |
⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∉ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ 𝐵)𝜑) | ||
Theorem | complss 3751 | Complementation reverses inclusion. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by BJ, 19-Mar-2021.) |
⊢ (𝐴 ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴)) | ||
Theorem | compleq 3752 | Two classes are equal if and only if their complements are equal. (Contributed by BJ, 19-Mar-2021.) |
⊢ (𝐴 = 𝐵 ↔ (V ∖ 𝐴) = (V ∖ 𝐵)) | ||
Theorem | elun 3753 | Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.) |
⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶)) | ||
Theorem | elunnel1 3754 | A member of a union that is not member of the first class, is member of the second class. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐶) | ||
Theorem | uneqri 3755* | Inference from membership to union. (Contributed by NM, 21-Jun-1993.) |
⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ⇒ ⊢ (𝐴 ∪ 𝐵) = 𝐶 | ||
Theorem | unidm 3756 | Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝐴 ∪ 𝐴) = 𝐴 | ||
Theorem | uncom 3757 | 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.) |
⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴) | ||
Theorem | equncom 3758 | If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 3758 was automatically derived from equncomVD 39104 using the tools program translatewithout_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.) |
⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵)) | ||
Theorem | equncomi 3759 | Inference form of equncom 3758. equncomi 3759 was automatically derived from equncomiVD 39105 using the tools program translatewithout_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.) |
⊢ 𝐴 = (𝐵 ∪ 𝐶) ⇒ ⊢ 𝐴 = (𝐶 ∪ 𝐵) | ||
Theorem | uneq1 3760 | Equality theorem for the union of two classes. (Contributed by NM, 15-Jul-1993.) |
⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) | ||
Theorem | uneq2 3761 | Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)) | ||
Theorem | uneq12 3762 | Equality theorem for the union of two classes. (Contributed by NM, 29-Mar-1998.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) | ||
Theorem | uneq1i 3763 | Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶) | ||
Theorem | uneq2i 3764 | Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵) | ||
Theorem | uneq12i 3765 | Equality inference for the union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷) | ||
Theorem | uneq1d 3766 | Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) | ||
Theorem | uneq2d 3767 | Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)) | ||
Theorem | uneq12d 3768 | Equality deduction for the union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) | ||
Theorem | nfun 3769 | Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵) | ||
Theorem | unass 3770 | 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.) |
⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵 ∪ 𝐶)) | ||
Theorem | un12 3771 | A rearrangement of union. (Contributed by NM, 12-Aug-2004.) |
⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = (𝐵 ∪ (𝐴 ∪ 𝐶)) | ||
Theorem | un23 3772 | A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ 𝐵) | ||
Theorem | un4 3773 | A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.) |
⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐷)) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐷)) | ||
Theorem | unundi 3774 | Union distributes over itself. (Contributed by NM, 17-Aug-2004.) |
⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶)) | ||
Theorem | unundir 3775 | Union distributes over itself. (Contributed by NM, 17-Aug-2004.) |
⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐶)) | ||
Theorem | ssun1 3776 | Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | ||
Theorem | ssun2 3777 | Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.) |
⊢ 𝐴 ⊆ (𝐵 ∪ 𝐴) | ||
Theorem | ssun3 3778 | Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶)) | ||
Theorem | ssun4 3779 | Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.) |
⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐶 ∪ 𝐵)) | ||
Theorem | elun1 3780 | Membership law for union of classes. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐵 ∪ 𝐶)) | ||
Theorem | elun2 3781 | Membership law for union of classes. (Contributed by NM, 30-Aug-1993.) |
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐶 ∪ 𝐵)) | ||
Theorem | unss1 3782 | Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶)) | ||
Theorem | ssequn1 3783 | 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.) |
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵) | ||
Theorem | unss2 3784 | Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∪ 𝐴) ⊆ (𝐶 ∪ 𝐵)) | ||
Theorem | unss12 3785 | Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷)) | ||
Theorem | ssequn2 3786 | A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.) |
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵) | ||
Theorem | unss 3787 | The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.) |
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶) | ||
Theorem | unssi 3788 | An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 𝐴 ⊆ 𝐶 & ⊢ 𝐵 ⊆ 𝐶 ⇒ ⊢ (𝐴 ∪ 𝐵) ⊆ 𝐶 | ||
Theorem | unssd 3789 | A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐶) & ⊢ (𝜑 → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) | ||
Theorem | unssad 3790 | If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐴. One-way deduction form of unss 3787. Partial converse of unssd 3789. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | ||
Theorem | unssbd 3791 | If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐵. One-way deduction form of unss 3787. Partial converse of unssd 3789. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | ||
Theorem | ssun 3792 | A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.) |
⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶)) | ||
Theorem | rexun 3793 | Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.) |
⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | ralunb 3794 | Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ (∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑)) | ||
Theorem | ralun 3795 | Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑) | ||
Theorem | elin 3796 | Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.) |
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | ||
Theorem | elini 3797 | Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ 𝐴 ∈ 𝐶 ⇒ ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶) | ||
Theorem | elind 3798 | Deduce membership in an intersection of two classes. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) | ||
Theorem | elinel1 3799 | Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐵) | ||
Theorem | elinel2 3800 | Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐶) |
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